LECTURES ON RANDOM MEASURES

by

Olav Kallenberg

Univepsity of ~tebopg, SWeden

and

Univepsity of Nopth Capolina at Chapel HillDepartment of Statistias

Institute of Statistics Mimeo Series No. 963

November, 1974

LECTURES ON RANDOM MEASURESby

Olav Kallenberg

PREFACE

Random measure theory is a new and rapidly growing branch of

probabi li ty of increasing interest both in theory and applications.

Loosely speaking, it is concerned with random quantities which can

only take non-negative values, such as e.g. the number of random variables

in a given sequence possessing a certain property, the time spent by a

random process in a certain region etc. In the special case when these

quantities are integer valued, our random measures reduce to point processes,

which constitute an important subclass of random measures. However, I am

convinced that general random measures are potentially just as important

from the point of view of applications. (See e.g. Kallenberg (1973c)

for some results supporting this opinion.)

The justification of random measure theory as a separate topic lies

in the overwhelming amount of important results which are specific to

random measures in the sense that they do not carryover to general random

processes. As a contrast, comparatively few results are specific in this

sense to the subclass of point processes. Indeed, it will he clear from

the development on the subsequent pages that a substantial part of point

process theory, as presented e.g. in the monograph of Kerstan, Matthes

and Mecke (1974), carries over to the class of general random measures.

..

The latter may therefore be considered as a natural scope of a theory.

These lectures contain research supported in part hy the Office of Naval eResearch under Contract NOOO1tl-67-I\-O~21-0002.

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The origin of these notes is a series of lectures delivered at the

Statistics Department at Chapel Hill during the spring semester of 1974,

where I intended to collect the (in my opinion) central facts about random

measures from the scattered literature in the field. During the course of

my lectures, however, I often elaborated new ideas and approaches, and as

a result I found many improvements of known theorems and proofs, and also

a large number of new results. The present notes should therefore be

regarded both as a survey of known results and as an original contribution

to the research in the area.

Here follows a summary of the contents and an indication about

the nature of the most important novelties. Section 1 is introductory.

After having set up the general framework, we demonstrate how the basic

point processes may be defined by a simple mixing procedure. Here and

in subsequent sections, the Laplace transform plays the role of a powerful

general tool. In Section 2, the main purpose is to extend the classical

simplicity criteria for point processes into a general theory of atomic

properties for arbitrary random measures. Section 3 is devoted to the

question of uniqueness of random measure distributions. In particular,

it is shown that the distribution of a simple point process or diffuse

random measure ~ is uniquely determined by the set function ~t(B) =

= Ee-t~B for bounded Borel sets B, where t > 0 is arbitrary but fixed.

Note that, as t + 00, ~t(B) + P{~B = O} and our result turns formally

into a well-known one for point processes. In Section 4, we derive the

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corresponding convergence criteria. A novelty here is a relationship

between the notions of convergence based on the weak and vague topologies

respectively. The idea of tightness leads in Section 5 to a simple

approach to the existence theorems of Harris and others.

Sections 6 and 7 are devoted to infinite dlivisibility and the limit

theorems for null-arrays. From a simple representation of an infinitely

divisible random measure l:;, we are able in Section 6 to draw interesting

conclusions about the relationship between the properties of l:;, itself

and its canonical spectral measure; in Section 7 we show among other things

how the above-mentioned uniqueness theorem in Section 3 leads in particular

cases to improved convergence criteria for null-arrays. In Section 8 on

compounding and thinning of point processes, the known results for the

case when the compounding variables f3 tend to zero in distributionn

are accompanied by an analogeous theory for the complementary case when

no subsequence tends to zero.

Symmetrically distributed random measures are examined in Section

9, and in particular we dwell on the case when the random measures are

at the same time infinitely divisible. The peak of our treatment is

the characterization of the class of canonical measures which can arise

in the corresponding spectral representations. After a general discussion

of Palm distributions in Section 10, containing improvements of known

uniqueness and continuity results, we finally enter in Section 11 into

the fascinating but difficult subject of characterizations by factorization

and invariance properties of the Palm distributions. As for the factorization

e

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properties, we are able to give a very simple proof of the Kerstan­

Kummer-Matthes theorem, while a modification of the original proof

yields a radically strengthened version of this result.

I hope that the technicalities of the above presentat,ion, mainly

intended for specialists, will not discourage newcomers to the field.

Great efforts have been made to provide a smooth approach to the subject,

and wherever I have relied on "known" facts, detailed references are given.

Still some requisites may prove helpful, including some basic knowledge of

measures on locally compact spaces (to be found in Bauer (1972)), and of

convergence of probability measures (as presented in Billingsley (1968),

Chapter 1).

Finally, I would like to express my thanks to my audience in Chapel

Hill, and in particular to Professor M. R. Leadbetter, for their interest

in this project and for several corrections and improvements of my original

draft. I am also grateful to the Institute of Statistics there for allowing

me to include these lecture notes in their Mimeo Series, and to Mrs. Melinda

Beck for her careful typing.

G~teborg in August 1974

O.K.

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CONTENTS

1. Basic Concepts 6

2. Atomic Propert ies 18

3. Uniqueness ....................•..................... :;0

4. Convergence ..........•.......................•...... :;9

5. Existence 57

6. Null-Arrays and Infinite Divisibility ..........•.... 66

7. Further Results for Null-Arrays 87

8. Compound and Thinned Point Processes 99

9. Symmetrically Distrihuted Random Measures .....•.... 113

10. Palm Distributions 1:16

11. Characterizations Involving Palm Distributions 151

..

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1. BASIC CONCEPTS

We could confine ourselves to random measures defined on the real

line R or on some Euklidean space kEN = {1,2, ... J , but it

turns out that we can choose a much more general framework without any

essential changes in notation or proofs. Hence we consider instead an

abstract topological space S which is assumed to be locally compact,

second countable and Hausdorff, (see e.g. Simmons (1963)). By choosing

a suitable metric p , we can convert S into a separable and complete

metric space. This possibility is often useful, but it is important to

realize that our definitions or results never depend on the actual choice

of p •

Let B be the Borel a-algebra in S, i.e. the a-algebra generated

by the class of open sets, and let B be the ring of all bounded B-sets.

(By definition, a set B is bounded whenever its closure B is compact.

In general, this is stronger than metric boundedness.) Write F = F(S)

for the class of !-measurable functions f: S + R = [0 00) and let+ "

F = F (S) denote its subclass of continuous functions with compact support.c c

To simplify statements, we introduce three types of subclasses of B.

By a DC-Bemiring (D for dissecting, C for covering) we mean a semi-ring

I c B (see e.g. Kolmogorov and Fomin (1970) for the definition) with the

property that, given any B E Band £ > 0 , there exists some finite

covering of B by I-sets of maximal diameter < £ in some fixed metric.

A DC-ring is a ring with the same property. Note that such classes always

exist and may be chosen countable. (We may e.g. take the ring generated

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by some countable base in B.) On R, typical DC-semirings and -rings

consist of finite intervals and interval unions respectively, hence our

notation. Finally, we say that a class C c B is covering if any set

B E B can be covered by finitely many sets in C .

Note that the notions of DC-semiring and --ring are purely topological

and do not depend on the choice of metric p. In fact, suppose that p

and pI are two metrizations of S, and let B E: Band E: > 0 be

arbitrary.I ­

Choose a bounded open set G ~ B . (Note that any countable

base in B covers B, and that a finite subcDvering exists since B

is compact.) Since G is compact, the identity mapping of G onto itself

is uniformly continuous with respect to p' and p, and in particular we

may choose some E ' > 0 such that p(s,t) < E whenever s,t E: G with

pI (s, t) < E I . Thus any covering of B with p I -diameters <E" =

= E: I 1\ p (B, (;c) (Gc = S\G denoting the complement of (;) is included

in G and hence has p-diameters < E. It remains to verify that E" > 0

which follows easily from the facts that B is compact while G is open.

We now introduce the class M = M(S) of Borel measures ]J on (S, B)

which are locally finite, i.e. take finite values on B, and further the

subclass N = N(S) of Z -valued measures (Z = {O, 1, ... })+ +

Let M

and N be the smallest a-algebras in M and N respectively making

the mapping ]J +]JB of M or N into R measurable for each B E B+

Note that, by monotone convergence, the integral

I lIere and below, B denotes the closure of B.

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is then automatically measurable for any f E F. More generally, assuming

f E F to be bounded with compact support and defining the measure f~ by

(f~)B = JBf(S)~(dS) , B E B , it is seen that the mapping ~ ~ f~ is

measurable. In particular, the restriction B~ = lB~ of ~ to B is

measurable for every B E B (As usual, lB denotes the indicator of

the set B, being equal to 1 on B and 0 elsewhere.)

LEMMA 1.1. The a-aZgebras M in M and N in N are both generated

by the mappings ~ ~ ~f , f E F ,and aZso by the mappings ~ ~ ~I ,c

I E r ~ for any fixed DC-semiring reB.

Proof. The same argument applies to both M and N , so let us e.g.

consider M Let M' be the a-algebra in M generated by the mappings

f E Fc By the definition of M, we have to prove that the

mapping ~ ~ ~B is M'-measurable for every B E B. If B is compact,

we may choose open bounded sets Gl ,G2, ... such that G + Bn By Urysohn's

lemma (cf. Simmons (1963)) there exist functions f l ,f2, ... E Fc such that

n EN, so for any ~ EM,

~B = ~lB ~ ~fn $ ~lG = ~Gn + ~B .n

Hence ~f + ~B , and the M'-measurability of ~B follows from that ofn

~f for n E Nn

We next consider an arbitrary fixed compact set C and define

v = {B E B: ~ ~ ~(B n C) is M'-measurable} .

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Then V is clearly closed under monotone limits and proper differences

and it contains S. On the other hand, V contains the class C of

compact sets, which is closed under finite intersections. Hence by Dynkin's

2monotone class theorem (cf. Bauer (1974)), V contains the a-algebra a(C)

generated by C. But a(C) = B , and so Jl -+ Jl(B n C) is M'-measurable

for every B E B. Since C was an arbitrary compact set, this proves

that M' = M

The assertion involving I is proved by a similar argument, but

instead of f n we consider a finite covering of B by I-sets In1 , ... ,Ink

contained in Gn

disjoint, and then

Since I is a semiring, we may take the I .nJ

to be

JlB ~ Jl u Inj = I JlI . ~ JlG + JlB ,j j nJ n

proving that I· Jl 1 . -)- JlB This completes the proof.J nJ

LEMMA 1.2. N EM.

Proof. Let I c B be a countable DC-semiring, and let Jl be a fixed

measure in the set

M = {Jl E M: Jll E Z , I E I} .+

For any fixed finite union U of I-sets, we further define

V {B E B: Jl(B n U) E Z } .-I-

Then D lS closed under monotone limits and proper differences, and it

2ThoUI~h commonly ascribed to Oynkln, this theorem

Si erpinski, Fund. Math . ..!2 (l92R).

is actually due to

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contains S since U may be written as a disjoint union of I-sets.

On the other hand, it is easily seen that V contains I, and I being

closed under finite intersections, it follows by Oykin's theorem that

V ~. a(I) ~ B and we obtain ~(B n U) E Z+ for every B E B. But for

given B E B , we may always choose U such that U 0 B , so we get

~B E Z for all B E B , proving that ~ EN. Hence MeN , and the+

converse relation being obvious, we have in fact N = M Since clearly

M EM, this completes the proof.

Let us now consider any fixed probability space (n,A,p). By a

random measure or point process ~ on S we mean any measurable mapping

of (n,A,p) into (M,M) or (N,N) respectively. By Lemma 1.2, a point

process may alternatively be defined as an N-valued random measure. Conversely,

any a.s. (almost surely w.r.t. P) N-valued random measure coincides with a

point process except on a P-null set (i.e. on an n-set of P-measure 0) .

For these reasons, we shall make no difference in the sequel between point

processes and a.s. N-valued random measures. Similarly, we shall allow a

random measure to take values outside M on a P-null set.

The class of random measures is clearly closed under addition and

multiplication by non-negative random variables. For a proof, it suffices

by the definition of M to observe that, if ~ and n are random measures

while a and S are R+-valued random variables, the quantity a~B + SnB

is a random variable for every B E B. Slightly less obvious is the

following

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LEMMA 1.3. A series L- t;. of random measures on S ~s itself a randomJ J

measure iff L.~.B < 00 a.s. for eachJ J

B E B .

Proof. First note that, by monotone convergence, ~ = L·~·J Jis (everywhere

on Si) a-additive on B and is therefore measure valued. Next suppose

that C c B is a countable base in S

outside some fixed P-null set A E A

Then ~C < 00 for all C E C

and since C is covering, it follows

that ~B < 00 B E B , except on A Hence ~ E M a.s. It remains to

prove that ~ is measurable, and by the definition of M , this follows

from the elementary fact that

B E B .

~B = L.~.B is a random variable for everyJ J

The distribution of a random measure or point process F, is the

probability measure p~-l on (M,M) or (N,N) defined by

M E M or N.

We further define the intensity E~ of ~ by

(EOB = E(~B) B E B ,

where E denotes the "expectation" or integral w.r.t. P. By Fubini's

theorem, E~ is always a measure, though it need not be locally finite.

We finally define the L-transform

L~(f) = Ee-F,f

of (L for Laplace) by

f E F .

Note in particular that, for any kEN and Bl , .. , ,'\ E B, the function

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LS[.r t.1B.] = E eXP[-.I t.~B.}J=1 J J J=1 J J

is the elementary L-transform of the random vector (sB1 , ... ,sBk)

One of the most important ways to define new random measures (or

rather their distributions) is by mixing. For a definition, let (8,T,Q)

be an arbitrary probability space and let {sS' S E 8} be a family of

random measures which mayor may not be defined on the same probability

space. If P{sS E M} is a T-measurab1e function of 8 for every M E M

we may then consider S as a random element in 8 with distribution Q

and form the mixture of with respect to Q , thus obtaining the

joint distribution R of the mixed random measure ~ and the mixing

element S. Formally, R is defined by the relation

is then a version of the conditional distribution of

(1 )

Note that

R(M x A) = JAP{~S E M} Q(dS)

Ps- 1S

M E M A E T .

given 8, and that

E'f(s,S) = E'E[f(~S'S) IS]

for any measurable function

for a complete discussion.)

f· M x 8 -+ R. + (See Lo~ve (1963) page 359

The measurability requirement above is not easy to verify. Here

is a cqnvenient criterion:

LEMMA 1.4. Let (8,T,Q) be an arbitrary probability space and let

{se ' 8 E 8} be a family of random measures on S. Then there exists

3 Here E' denotes expectation w.r.t. R .

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a unique probability measure R on M x T satisfying (1) if Lt.: (f)e

is T-measurable in e for every fixed f E Fc

Proof. The necessity of our condition follows by monotone convergence

from simple functions. Conversely, suppose that Lt.: (f) is measurablee

in e for every f E Fc

Then

L~ (L t.f.)e j J J

= E exp(-L t.t.: f.)j J (I J

is measurable in e for any fixed keN

Now it follows by the entended Stone-Weierstrass

theorem (see Simmons (1963)) that any continuous function g(x l , ,xk)

on which vanishes at infinity can be uniformly approximated by linear

combinations of the functions eXP(-L/jXj ) with t l , ... ,tk E R+

Therefore Eg(!;efl' ... '!;efk) is measurable in e for any such g ,

and hence, by an easy application of Urysohn's lemma, it follows that

the probability

is measurable in e for any t l , ... ,tk E R+

Let us now define V as the class of all sets M c M such that

P{[,e c M} is measurable. Then V is closed under proper differences

and monotone limits, and it contains M. On the other hand, it was shown

above that V contains the class C of all sets

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and since C is closed under finite intersections, it follows by Dynkin's

theorem that

f E Fc

But by Lemma 1.1, the right-hand side of this relation equals M, and

so P{~e E M} is indeed measurable in e for all M EM, as required.

We shall now introduce some important point processes and at the

same time illustrate the use of L-transforms and mixing.

For any s E 5 , let Os E N be defined by 0sB = lB(s) , B E B ,

and note that the mapping s -+ °Sis measurable B-+ M Hence is

a point process on 5 whenever , is a random element in (5,B) Writing

-1w = PT for the distribution of , , it is easily seen that 0, has

intensity -1E<\ = pT = W and L-transform

-f(T) -fE exp(-o,f) = Ee = we f E F .

Next suppose that n E Z+ and that T l' ... ,Tn

are independent random

elements in 5 with common distribution w E M Then any point processn

c;, which is distributed as L °T is called a sample process with intensityj=l j

nw. By independence, ~ has L-transform

-~fEe =n

-f nIT E exp(-o, f) = (we )

j =1 jf E F .

Here we may mix with respect to n = v , regarded as a random variable,

to obtain a mixed sample process with intensity Ev· wand L-transform

r - ~f E' ( -f) V ,I, ( -f)L:e =. we = If' we f E F ,

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where

S E [0,1] ,

is the (probability) generating function of v

In the particular case when v is Poissonian with mean a, we get

1jJ (s) -a(l-s)= e S E [0,1] ,

and the L-transform of ~ becomes

(2)-~f f fEe = exp(- a(l - we- )) = exp(- 1..(1 - e- )) f E F ,

where we have put A = aw = Et;. This measure A is totally bounded, hut

we may also define point processes ~ satisfying (2) when A E M is arbitrary

unbounded. For this purpose, let Sl,52 , ... E B be a disjoint partition of 4ItS. Then the restrictions S.A of A to S. are totally bounded, and so

J J

there exist independent point processes ~1'~2' on S with L-transforms

(3) ( -f )E exp(-E;.f) = exp -(S.A) (1 - e )J J

f E F .

Since their sum (= LjSj satisfies

E~B = L E~.B = L (S.A)B. J . JJ J

= (L S.A)Bj J

AB < 00 B E B ,

it follows by Lemma 1. 3 that t, is a point process. Furthermore, it 15

seen from (3) and the independence of t,l'sZ' ... that, for any f ( F ,

(' -r,f:e· =-Sof

E If e Jj

-Cf= JT Ec J

J

cxp ( -L(S . A) (l. JJ

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which shows that I; satisfies (2) . (The formal calculations here are easily

justified, using the fact that I;.f and (S.A) (1 -f non-negative.)- e ) areJ J

Any point process I; satisfying (2) will be called a Poisson process with

intensity A. If I; is a process of this type, it is easily seen from

(2) that, for any kEN and disjoint Bl , ... ,Bk E B , the random variables

I;B l , ... ,I;Bk are independent and Poissonian with means ABl , ... ,ABk .

Let us now consider a Poisson process I;a with intensity aA , where

a E R+ while ~ EM. By Lemma 1.4, we may then consider a as a random

variable and mix with respect to its distribution to obtain a mixed Poisson

process I; with L-transform

where

-E;f ( -f )Ee = E exp -aA(l - e ) f E F ,

Ht)-at= Ee t 2: 0 ,

is the L-transform of a. More generally, it is possible by Lemma 1.4

to consider A = n in (2) as a random measure, thus obtaining a Cox process

directed by n, possessing the L-transform

(4) f E F .

kLet us finally consider some fixed measure l.l E N , say l.l = I <s

j=l t j

where k E Z = Z u {oo} and t. E S , j ~ k Further suppose that+ + J

are independent and identically distributed R+-valued random

variables, say with LS

= ~ . Then it follows from Lemma 1.3 that

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is a random measure on S , and we get for f E F

IT E exp(-S.f(t.)) = IT ~ 0 f(t.)j J J j J

~ = 1:.S.<\J J j

Ee -H = E exp(- 1: S.f(t.)) =. J JJ

= exp(I log ~ 0 f(t.)) =j J

exp(]..I log ~ 0 f)

By Lemma 1.4, it is permissible to mn with respect to ]..I

as a point process, thus obtaining

n when regarded

f E F .

Any random measure with this mixed distribution will be called a (j-compound

of n. In the particular case when S attains the values a and 1

only with probabilities 1 - P and p respectively, ~ will be called

a p-thinning of n. In this case, (5) takes the form

(6) f E F .

Scrutinizing the arguments used in the above constructions, it is

seen that they do not depend on the nature of S. Thus we may e.g. proceed

as above to construct Poisson processes with arbitrary a-finite intensities

on any measurable space (s,B) .

NOTES. !Iere and below, our main source is Kallenberg (1973a). However,

Lemma].4 is new and is essentially a strengthening of proposition 1.6.2

in Kerstan, Matthes and Mecke (1974). The present method on constructing

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Poisson and Cox processes is simpler and more general than methods in

the current literature.

PROBLEMS.

1.1. By a o-ring ReB we mean a ring which is closed under non-decreasing

bounded limits. Show that, if 1 is a DC-semiring, then B is the

smallest a-ring generated by 1 In symbols: B = a(l) .

1.2. Show that Lemma 1.1 remains true for any semiring 1 c B with

a(I) = B.

1. 3. Let 1 c B be an arbitrary DC-semiring. Show that Lemma 1.4 remains

true with F replaced by the class of all functions of the form

L~=lc

LI I for arbitrary k E N , t l , ,tk E R andJ . +

JII' ,Ik E 1

1.4. Let ~ be a Cox process on S directed by n Show that ~

has independent increments, in the sense that ~Bl' ... ,~Bk are

independent for any kEN and disjoint Bl , ... ,Bk E B , iff

this is true for n Prove the corresponding fact for compound

point processes with 8 ~ O. (Hint: use analytic continuation.)

2. ATOMIC PROPERTIES

Let Md denote the class of diffuse (or non-atomic) measures in M.

Every measure ~ E M may clearly be written in the form

(1)k

~ = ~d + L b.o tj =1 J j

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for some ~d E Md , k E Z+ = Z+ U {~} , bl , b2 , ... E R~ = (O,~) and

distinct t l , t 2 , ... E S , and this decomposition is clearly unique apart

from the order of terms. Moreover, 11 E N iff " - 0"d - and

When considering random measures, we often need a measu~able decomposition.

LEMMA 2.1. There exists a decomposition (1) of' every 11 E M such that

the quantities 11d E Md , k E Z+

~e measurable functions of 11

The measurable functions occurring in Lemn~ 2.1 are by no means unique,

not even apart from the order of terms. However, when considering random

measures ~ on S with ~S < ~ a.s. (which holds automatically when

S is compact), we may require the atom sizes 81 , 82, ... to be taken

in order 81 ~ 82 ~ ... and the corresponding atom positions Tl , T2' ...

to be chosen in random order within sets of e~lal 8j . (We shall always

assume the basic probability space (rl,A,P) to be rich enough to support

any such randomization.) Though the decomposition

(2)v

~ = ~d + Lj =1

8·0J T .J

is still not unique unless all the atom sizes are different with probability

1 , we obtain in this way a unique joint distribution of the random elements

occurring in (2).

Par the proof of Lemma 2.1, we shall need an auxiliary result, the

formulation and proof of which requires some further terminology and notation. _

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denotes the diameterI. I

{B .} c B is a null-ar~y ofnJ

n E N form a finite disjoint

(Note that the last condition is

B E B , we shall say that

B

Given any set

par' ti tions of if the B. for fixednJ

partition of B and if max. 1 B 01 -+ 0 , whereJ nJ

in any fixed metrization p of S

independent of the choice of P.) For any p E M and £ > 0 , we define

*pEN and pi E M by£ £

* Lp B ::: I (p) B E B ,£ sEB {p{shd

p'B ::: pB - L lJ{s}l{lJ{s}~d (ll) B E B •£ sEB

LEMMA 2.2. Let II EM, £ > 0 and B E B , and let {B o} c B be anJ

null-array of partitions of B. Then

lim L I (p)J

o {lJB 0 ~e:}n-+<x> nJ*::: p Be:

PROOF. For sufficiently large n EN, all ll-atoms in B of size ~ £

will lie in different partitioning sets Bnj , and we get

L I (ll)o {pB .~e:}J nJ

*~ p Be:

Thus it remains to prove that

(2)

If (2) were false, there would exist some subsequence N' c N and some

numbers jn' n E N' , such that

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(3) n E N' .

Choosing arbitrary n E N' , there exists by compactness

some further subsequence Nil c N' and some s E B such that sn -+ s ,

n E Nil , and sin¢e lB. I -+ 0 , it follows from (3) that ll'G ~ E fornJn £

every open set G E B containing s But then ll'{S} ~ e: , which con-E

tradicts the definition of II ' Hence (2) must be true, and the lemma£

is proved.

PROOF OF LEMMA 2.1. Define the set function ~ = ~(ll) by

(4) *= II B£

B E B E > 0 .

By the Caratheodory extension theorem, (4) defines for each II E M a

unique measure ~ on S x R: ' and it is easily seen that ~ is purely

atomic with all its atoms of unit size. Furthermore, it follows from

for some measurable

Lemmas 1.1 and 2.2 that ~

that the lemma is true for

is a measurable function ofk

~ , i.e. that ~ = I 0j =1 cr j

II Suppose

functions k E Z and cr l , °2' E S x R' Then (1) holds with+ +

cr. - (t . , b. ) , and since t. and b. are measurable functions of theJ J J J J

pair (t. , b. ) , the measurability in (1) will follow by Lemma 1.3. WeJ J

may therefore assume from now on that lld = 0 and bl :: b2 - - 1

Let s1' s2' ... be an arbitrary dense sequence in S and note

that p (s, sn) = p (t, sn) , n E N , implies s = t (To see this, let

N' c N be such that s -+ s n E N' , and note that thenn

-22-

p(S,t) 5 peS, s ) + pes, t) = 2 pes, s ) + 0n n n n € N' .)

We may therefore introduce a linear ordering of S by writing s < t

whenever, for some kEN,

pes, s.) = pet, s.)J J

j = 1, ... ,k - 1

For any fixed disjoint partition Bl , B2, '" E B of S, we now order

the atom positions t.J

of ~ , first according to their occurrance in

Bl , B2, ... , and then, within each Bn , according to their linear order.

We have to show that t l , t 2 , ... are measurable when ordered in

this way, and by induction it suffices to consider t l ' i.e. to prove

that the set {t l € B} is measurable for every fixed ~ E B. Now this

set is clearly expressible by means of countable set operations in terms

{p (c n r}) = k} cof the sets B. n {s: p(s, s. ) < with C = B or B

1 J

and with arbitrary i,j E N , kE Z+ and rational r > 0 Since

the latter sets are measurable, this completes the proof.

We shall now establish a simple relationship between the one-dimensional

distributions of a random measure ~ and the intensities of the corresponding

*point processes ~£'

o < a < b 5 00 •

* * *£ > 0 , or more generally of ~[a,b) = ~a - ~b '

THEOREM 2.3. Let ~ be a random measure on S, let B E B and let

{B .} c B be a null-array of partitions of B. Further suppose thatnJ

o < a < b s; 00 Then

(5 )

-23-

*E~[a,b)B = lim L P{~B . e [a,b)}n-+-<o j nJ

holds provided Et;'Ba< co , while in general

(6) 1iminf? P{t;Bnj e [a,b)} .n-+-<o J

PROOF. By taking differences in Lemma 2.2, we get

(7) lim? l{t;B .e[a b)}n-+-<o J nJ '

and hence, by Fatou's lemma,

1iminf E ? l{~B .era b)}n-+-<o J nJ '

= 1iminf L P{~B . E [a,b)} ,n~ j nJ

*proving (6). Since (6) implies (5) in the case E~[a,b) = co , it remains

*to prove (5) in the case when E~Ia,b)B and E~'B are both finite.a But

then (5) follows from (7) by dominated convergence, since we have

and hence

I} u {~'B .a nJ

~ a} ,

* \ -1~ L ~I b)B . + L a ~'Bh··j a, nJ j a J

~* B a- 1nB- ~[a,b) + sa

We next give a simple criterion ensuring t; to have no atoms of

size ~ £. Given any covering class C c B and any a > 0 we shall

-24-

say that ~ is a-regu~ar w.r.t. C, if there exists for every fixed

C E: C some array {C .} cC of finite coverings of CnJ (one for each

n E: N ), such that

(8) lim LP{~C . ~ a} = 0 .. nJn+oo J

THEOREM 2.4. Let ~ be a random measure on S, ~et C c B be a covering

cZass and ~et a > 0 *Then ~ = 0 a.s. if ~ is a-reguZar w.r.t. C.a

The converse is aZso true provided C is a DC-semiring and E~ E: M.

PROOF. Suppose that ~ is a-regular w.r.t. C, let C E: C be arbitrary

and let {C .} c C be an array of coverings of C such that (8) holds.nJ

Then

p{max ~{s} ~ a} S P{max ~C . ~ a} = P U {~C . ~ a} S I P{~C . ~ a} ,s C . nJ . nJ . nJ

J J J

*and it follows from (8) that ~ C = 0 a.s. Since C is covering, thisa*implies ~a = 0 a.s.

*Conversely, let E~ E: M and ~a = 0 a.s. , and suppose that C is

a DC-semiring. For every C E: C we may then choose a nUll-array {C .} c CnJ

of partitions of C , and we get by Theorem 2.3

*lim I P{~C . ~ a} + E~ C = 0 ,. nJ an+oo J

proving a-regularity of ~ w.r.t. C.

-25-

It is often useful to replace the global regularity condition (8)

by a local one. For a first step, note that if C is a DC-semiring and

if A E M is arbitrary, then (8) is implied by

(9) lim SUP{A~ P{~C ~ a}: C E C£+0

We shall show that the uniform convergence in (9) may essentially be replaced

by pointwise convergence.

THEOREM 2.5. Let ~ be a ~ndom measupe on S , let A E M and let

*a > O. Then ~a = 0 a.s. if

(10) lim sUP{Ai P{~I ~ a}: I E I£+0

sET S E S ,

fop some DC-semiping I c B 3 and also if

(11) lim sup{~~ P{~C ~ a}: C E C£+0

SEC S E S ,

whepe C is the class of closed or the class of open B-sets.

PROOF. Suppose that *P{~ ~ a} > a , and leta I c B be a DC-semiring.

Since I is covering, we may choose some *I E I such that Pis I > O} > aa

We may further choose some null-array {I.} c I of partitions of I , andnJ

we may assume {I.} to be nested in the sense that it proceeds by successivenJ

refinements. Then

(12) *Pis I > O}a* *= P U {~ II' > O} ~ I P{~ II' > a} ,

j a J j a J

and it follows that

(13)

-26-

1 *max xr-- pis II' > a} ~j lj a J

provided % is interpreted as a In fact, this follows trivially

from (12) if AI = a , while if AI > 0 , insertion of the converse of

(13) into (12) yields the contradiction

*P{s I > a} .a

By (13) there exists an index jl such that I = II' satisfies1 J l

1 * 1 *All P{sall > a} ~ rrP{sa l > O}

and proceeding inductively, we may construct a sequence 11 J 12 J •• ,

in I with lIn' ~ 0 and such that

1-- P{OAI n

nn EN.

Now {I} has a non-empty intersection {s}, and the last relation showsn

that (10) is violated at s.

Let us now introduce the class

I = {B E B: AaB = a} ,

where aB = B n BC

denotes the boundary of B If we can show that I

is a DC-ring, it follows that we may apply the first assertion to I.

Now the ring property of I follows from the elementary relations

a(B u C) c aB u ac a(B n C) c aB u ac

To see that I has the DC-property, it suffices to show that, for any

fixed t E S , the ball

-27-

S(t,r) = {s E S: p(s,t) < r}

belongs to I for arbitrarily small r > 0 For a proof, note first

that S(t,r) E B for small r, say for r < £ , since S is locally

compact. Next verify that

as(t,r) c is E s: p(s,t) = r} ,

and conclude that AaS(t,r) = 0 and hence S(t,r) E I for all but countably

many r < £ •

With the above choice of I we have AI = AI I E I , and hence

1IT p{O ~ a} I E I ,

which proves that (11) implies (10) when C is the class of closed sets.

In the case of open sets, let £ > 0 be arbitrary and approximate each

I E I by an open bounded set G :J I such that IGI < 2111 v £ and

rAJ AI > 0 ,AG ~

P{t;I ~ a} AI = 0

Then

1IGP{~G ~ { 2l~I P{~I ~ a}

a} ~

AI > 0 ,

AI = 0 ,

and again (11) implies (10).

...28-

From Theorem 2.4 it follows in particular that, given any covering

class C, a random measure ~ is a.s. diffuse provided ~ is a-regular

w.r.t. C for every a > 0 In this case we shall say, simply, that ~

is regular w.r.t. C

2.5.

Other diffuseness criteria are implicit in Theorem

Another case of particular interest is that of point processes. If

is a point process, we write *= ~1and say that is simple if

*~ ~ a.s. Since for point processes ~i = 0 a.s., (5) holds generally

with a = 1 Furthermore, we obtain convenient simplicity criteria by

taking a = 2 in Theorems 2.4 and 2.5.

We conclude this section with a simple but useful lemma.

LEMMA 2.6; Let ~ be a point process on S and let I c B be a DC-

semiring. Then ~ is simple iff

(14) *P{~I > I} ~ P{~ I > I} I ~ I .

PROOF. The necessity of (14) is obvious. Conversely, suppose that (14)

holds, and note that then

* *(15) 0 ~ P{~I > ~ I = I} = P{~I > I} - P{~ I > I} ~ 0 I E I .

Letting I E I be fixed and choosing a null-array {I.} c InJ

of partitions

of I , we obtain by (15) and Lemma 2.2

-29-

* * *P{sI > s I} = P U {sI . > ~ I .} = P U {sI . > s I . > I}j nJ nJ j nJ nJ

covering, it followsproving that

s P U {s*I . > I} = p{I 1 * >j nJ j {s I .>0

nJ*(s ~)I = 0 a.s. Since I is

*that ~ - s = 0 a.s., as asserted.

NOTES. In the point process case, Theorem 2.3 with a = 1 and b = 00

is known as Korolyuk' s theorem, while the converse part of Theorem 2.4

with a = 2 is known as Oobrushin's lemma, see Leadbetter (1972) and Belyaev

(1969). The present generalizations to random measures are new, although a

first ~tep towards Theorem 2.4 was given in Kallenberg (1975) while the

ideas behind Theorem 2.5 are implicit in Kallenberg (1973d). See also

Gikhman and Skorokhod (1969) for a version for random processes in 0[0,1] ~of the direct part of Theorem 2.4. We further point out that our conditions

(10) and (11) generalize the notion of analytic: orderliness in Daley (1974),

while (9) generalizes his notion of uniform analytic orderliness. Finally,

we mention that Lemma 2. I was given without proof in Kallenberg (l973b)

while Lemma 2.6 is taken from Kallenberg (1973a).

PROBLEMS.

*2.1. The measurability of ).l -+).l , a > 0 , is an immediate consequencea

of Lemma 2.1. Give a direct argument, proving the measurability

of ).l -+ ).l~ a > 0 and conclude that ).l -+).ld is measurable.

(All these facts are of course contained in Lemma 2.1.)

2.2. Prove that the events {s = ~d} and (in the point process case)*{t, = ~} are measurable. (Hint: considE~r the differences.)

-30-

2.3. Prove that the notion of null-array of partitions and the conditions

(10) and (11) are independent of the choice of metric p •

2.4. Let s be a Cox process directed by n. Show that s is simple

iff n is diffuse. Prove also that, if s is a p-thinning of

n (p>O) , then sand n are simultaneously simple. (Hint: consider

first the case of non-random n .)

2.5. Show that Lemma 2.6 follows easily from Theorem 2.3 in the particular

case when E~2 € M .

2.6. Let s be a random measure on S, let 1 c B be a DC-semiring*and let a > O. Prove that ~ = 0 a.s. iff P{sI ~ a} ~ P{s'I ~ a} ,a a

I € 1 .

2.7. Show by an example that the decomposition in Lemma 2.1 is not unique

in general, not even apart from the order of terms. (Hint: take

e.g. S = [0,1] , v = 2 Bl = B2 = 1 , and choose (T l , T2)

in two different ways.)

2.8. Show that the converse of Theorem 2.4 may be false when F~ ~ M .

(Hint: consider a suitable mixture w.r.t. n E N of the non-random

simple point processes on R with atoms at j2-n j EN.)

3. UNIQUENESS

The uniqueness results of this section will playa basic role in

the subsequent discussion of convergence in distribution.

with a theorem which applies to arbitrary random measures.

and V = (Bl , ... ,Bk) € Bk , kEN , we abbreviate (llBl ,

Let us start

For any p E M

Write d

-31-

for equality in distribution, i.e. d~, Tl iff

1 -1P~- = Pn .

THEOREM 3.1. Let E; be a random measure on S. Then PE;-l is uniquely

determined by PCE;f)-l, f E Fc ' and even by L~(f) , f E Fc It is

further determined by p(~V)-l , V E rk , kEN, for any fixed DC-semiring

reB.

PROOF. Let and be two random measures on S satisfying dE; n E;V = nV ,

V E rk k E N , for some fixed DC-semiring J c B . Letting V denote

the class of all sets M E M such that PE;-lM = Pn -1M , it is seen that

V contains M and is closed under proper differences and monotone limits.

Furthermore, V contains by assumption the class C of all sets of the

form

k f> N t l , ••. , t k E R+

Since this class is closed under finite intersE!ctions, it follows by Dynkin' s

theorem that V:J o(C) . But o(C) = M by Lemma 1.1, and we reach the, d

desired conclusion E; = n .

A similar argument shows that p~-l is de!termined by the class of

-1all distributions P(t;f l' ... , E;fk) kEN, f l' ... ,fk E Fc

But by the uniqueness theorem for mUlti-dimensional L-transforms,

-1,~fk) is in turn determined by

L f f (tl' ... , t k )f;, l'H.,f;, k

where f = \.t.f.LJ J J

Since f E F , this completes the proof of the firstc

assertion.

-32-

Note in particular that the L-transforms which were calculated in

Section 1 can be used to define the corresponding basic point processes.

COROLLARY 3.2. Let n be a random measure on S and let t;, be a Cox

directed by Then -1 is determined by pt;,-l This statementprocess n • Pn

is also true when n is a point process on S and t;, is a B-compound of

n "' provided PB- 1 is known and such that B ~ o .

PROOF. Suppose that t;, is a Cox process directed by n Solving for

f in 1 - e-f obtain from (1.4) for any fixed f F, we E c

(1)

where Ilfll = sup f(s) Now Lnf is analytic in the right half-plane,

and so it is completely determined by its values on any finite interval.

Hence by (1), is determined by pt;,-l , and since f E F wasc

arbitrary, the assertion follows from Theorem 3.1. A similar argument

applies to the case of compound point processes, except that (1) is then

replaced by

o ~ t < -I If11-1 log P{B = O} ,

where -1cP is the unique inverse of cP on the interval (p{ 8 = a} , 1] .

Stronger results than in Theorem 3.1 are attainable if we confine

ourselves to simple point processes or diffuse random measures.

-33-

THEOREM 3.3. Let ~ and n be two point proeesses on S" and suppose

that ~ is simp le. Further suppose that U c B is a DC-ring and I c B

a DC-semiring. Then d *~ = n iff

(2) P{~U = O} = P{nU = O} U E U ,

and we have even

(3)

d~ = n provided in addition

P{~I > I} ~ P{nI > I} I E I .

PROOF. The conditions (2) and (3) are clearly necessary. Conversely,

suppose that (2) holds. As before, the class V of all sets MEN

-1 -1satisfying p~ M = Pn M is closed under proper differences and monotone

limits, and it contains N By assumption, it: also contains the class C

of all sets of the form {~E N: ~U = O}, U E U. Now it is easily seen

that

{~U = O} n {~V = O} = {~CU u V) = O} u,V E U ,

and since U is closed under finite unions, it follows that C is closed

under finite intersections. Hence, by Dynkin's theorem, V:::> aCC). But

*from Lemma 2.2 with e: = I it is seen that the: mapping ~ -+ ~ is measurable

w. r. t. a (C) * d *so we get ~ = t;, = n , proving the first assertion. If in

addition (3) holds, then

*Pin I > I} = P{~I > I} ~ P{nI > I} I E I ,

and it follows by Lemma 2.6 that * dn = n = t;, • This completes the proof.

-34-

Note that a variety of related results may be obtained by using the

simplicity criteria of Section 2 rather than condition (3). A corresponding

remark applies to the next theorem.

THEOREM 3.4. Let ~ and n be ~o point ppocesses (pandom measupes)

on S, and suppose that ~ is simple (diffuse). Fupthep suppose that

U c B is a DC-ping and C c B a coveping class, and that sand t

ape fixed peal numbeps with 0 < s < t. Then ~ g n iff n is simple

(a.s. diffuse) and

(4) U E U ,

and also iff (4) holds and in addition

(5) E -s~C < E -snCe _ e C E C .

In the case E~ EM, we may peplace (5) by

(6 ) E~C 2': EnC C E C .

It is interesting to observe that (2) is a limiting case of (4) obtained

by letting t ~ 00 •

PROOF. Let us first consider the point process case, put p = 1-t

- e~

and let ~ and n be p-thinnings of ~ and n respectively. By (1.6)

we get

B E B

-35-

and letting x -+- (X) , we obtain by monotone convergence

(8) ~ ( ) -·t~BP{~B = O} = E exp ~B log(l - p) = Ee B E B .

By (4), (8) and the corresponding relation for n , we obtain

P{~U = O} = Ee-t~U = Ee-tnU = P{nU = O} U E U ,

and it follows from Theorem 3.3 that~* d ~*

t;, = n Now and are

and n (cf. Problemsimultaneously simple and correspondingly for

~ d ~*2.4), so we get ~ = n in general, and if n

n

is simple even~ d ~

t;, = n .

Thus the first assertion follows by Corollary ~;. 2.

To prove the second assertion, we may assume that

(5) holds. Choosing

~ d ~*

t;, = n and that

in (7) and in the corresponding relation for n , which is possible since

s < t , we obtain from (5)

~* ~ ~

Ee-xn C = Ee-x~C ~ Ee-xnC

or equivalently

C E C ,

C E C .

But here the random variable within brackets is non-negative, so it must~ ~*

in fact be a.s. 0, and it follows that nC = n C a.s., C E C since

-36-

~

x > O. Writing this result in the form (n~*

n )C = a a.s., C € C ,

and using the fact that C is covering, we obtain n = n a. s., so again~ d ~

~ = n , as required. The last assertion follows by a similar argument,~

based on the relations E~ = pE~ and En = pEn, which may e.g. be obtained

from (7) by differentiation.

The random measure case may be proved from Theorem 3.3 by a similar~

argument where we consider Cox processes ~ and n directed by t~

and tn respectively. Alternatively, we may choose any fixed x > t~

let ~ and n be Cox processes directed by x~ and xn respectively,

and apply the point process case of the present theorem.

As an application, we prove the following partial strengthening of

Corollary 3.2.

COROLLARY 3.5. Let ~ be a S-compound of n and suppose that

p = P{S > O} > a . Further suppose that Cn is known to be simple for

C E B with nC ~ a . Then -1 and PS-l are uniquely determinedsome Pn

by p~-l and p

PROOF. By (1.5),

(9) Ee-t~B = E exp(nB log ~(t)) B E B

and letting t 7 ~ , we obtain by monotone convergence

P{~B = a} = E exp(nB 10g(1 - p)) B E B .

-37-

Considering sets B contained in C , it follows by Theorem 3.4 that

P(Cn)-l is uniquely determined by PI;-l. Let us now write (9) for

B = C in the form

(10) LI;C = LnC 0 (-log <p) •

Since nC ~ 0 by assumption, the function L (~n... is strictly increasing,

so it must have a unique inverse

and we obtain from (10)

on its ,range (P{nC = Ol, 1] ,

proving that <p is determined by PI;-l. We may now apply Corollary

3.2 to complete the proof.

NOTES. In Theorem 3.1, the first assertion is due to Prokhorov (1961),

(see also von Waldenfels (1968) and Harris (1971)), while the second assertion

is essentially a standard result for random prbcesses. Theorem 3.3 was first

proved in the particular case of Poisson processes by R~nyi (1967), and then

in the general case, independently by Mt5nch (1971) (in a weaker formulation)

and Kallenberg (1973a). As for Theorem 3.4, the first assertion was proved

for diffuse random measures (again in a weaker formulation) by Mt5nch (1970),

and independently by Ka1lenberg (1973e) and Grande1l (1973). The point

process case of that theorem is new as are the last two assertions. Finally,

Corollary 3.2 is due in the thinning case to MElcke (1968) and in the case of

Cox processes to Kummer and Matthes (1970), while the assertions for compound

processes in Corollaries 3.2 and 3.5 are new.

-38-

PROBLEMS.

3.1. Show that, for any random measure ~,the distribution of ~ is

determined by L~(f) for all linear combinations f of indicators

IT' lEI, for any fixed DC-semiring 1.

3.2. Theorem 3.1 clearly extends to arbitrary semirings generating B.

Prove that the first assertion in Theorem 3.3 extends to arbitrary

rings U generating B. (Hint: Show that the class of sets U

satisfying (1) is closed under monotone bounded limits, and hence,

by the classical monotone class theorem in Lo~ve (1963), must contain

B = a(U) . Then apply Theorem 3.3 with U = B.) Even the second

part of Theorem 3.3 admits a similar strengthening, cf. Kerstan,

Matthes and Mecke (1974), page 43.

3.3. Prove an alternative to the second part of Theorem 3.4 by applying

the same method involving Cox processes to the second part of Theorem

3.3.

3.4. Prove that the distribution of an arbitrary point process ~ is

not even determined by P(E,;B)-l , B E B (Hint: consider two

and2 d andrandom vectors a 13 on {O,1,2} such that a ~ 13 ,

still a l ~ 13 a 2 ~ 13 and a l + a 2~ 13 + 13 2

.)1 2 - 1

3.5. Show that the ring property of U in Theorem 3.3 is essential.

(llint: we may e.g. introduce a suitable dependence in a stationary

renewal process.) Cf. Kerstan, Matthes and Mecke (1974), page

213, for a stronger result and further references.

3.6. Prove that, for an arbitrary random measure ~,the intensity

E~ determines p~-l iff E~ = a .

3.7. Show that

-39-

as t-l-O, B E B . This is

interesting since, for simple point processes and diffuse random

measures ~, p~-l is determined by the left-hand sides for arbitrarily

small t > 0 , and still it is not determined by the limit.

3.8. Assume that S is countable. Show that there exists some fixed

function f E F such that p~-l is determined by P(~f)-l for

every point process ~ on

numbers f(s) , s E S , be

above and below by positive

S with ~S <: 00 a.s. (Hint: let the

rationally independent and bounded

constants.)

3.9. Show that, if the class C in Theorem 3.4 is a DC-semiring, then

~ may be allowed to have atoms of fixed size and location. (Cf.

the proof of Theorem 7.8 below.)

4. CONVERGENCE

In view of Theorem 3.1, it appears natural to say that the random

measures

whenever

/;n' n EN, converge in distribution to ~ (wri tten

(1) ~ V ~ ~Vn

kEN ,

for some fixed DC-semiring I, and this is also the mode of convergence

used in much classical work (with I the class of real intervals). However,

in the sense of (1), (cf. the trouble

(1) requires too much in general since

does not necessarily imply o ~ 0T T

n

dT -l- Tn

for random elements in s

with pointwise convergence of distribution functions). Moreover, the above

-40-

mode of convergence depends strongly on the choice of I As will be

seen below, we can avoid these disadvantages by assuming that I satisfies

sal = 0 a.s., lEI. We prove that OC-semirings I with this property

do exist. Let us write B~ = {B E B: ~aB = 0 a.s.}

LEMMA 4.1. Bs is a DC-ring for every random measure ~ on S.

PROOF. Proceeding as in the proof of Theorem 2.5, we need only show that

(2) ~{s E S: p(s,t) = r} = 0 a.s.

for any fixed t E S and arbitrarily small r > O. For this purpose,

let s > 0 be such that

Set, s+) = {s E S: p(s,t) ::; dEB,

and define the random process X in O[O,s] by

Then

X(r) = ~S(t, r+) r E [O,s] .

~{s E S: p(s,t) = r} = X(r) - X(r-) r E (O,s] ,

and since we can have P{X(r) ~ X(r-)} > 0 for at most countably many

r E (D,s] (cf. Billingsley (1968), page 124), it follows that (2) holds

almost everywhere on [O,s] This completes the proof.

-41-

To be able to apply the powerful theory of weak convergence of pro-

bability measures, we shall now introduce a metrizable topology in the

measure space such that (1) subject to the above restriction on I is

equivalent to convergence in distribution in the sense of this theory.

For any

to )J

11, ]11' 11 2 , •••

vand write ~ ~ ~n

in M or

whenever

N , we shall say that l1n

11 f ~)Jf for all f E Fn c

tends vaguely

The so

induced vague topology in M or N is known to be metrizable and separable

(see Bauer (1972); it is indeed even Polish, see Lemma 10.1 below). It is

often useful to relate the notion of vague convergence to the better known

notion of weak convergence. For any 11, 11 1, )J2' ... E M with l1S < 00

we say that l1 n tends weakly to lJ and write whenever 11 f ~ l1fn

for every bounded continuous function f E F The reader should verify

that wIln -+ 11 iff v

11 ~)Jn and )J S ~ )JS < 00

n

The next lemma shows that the random elements in the topological

Borel spaces M and N coincide with our random measures and point processes

respectively.

LEMMA 4.2. M and N coincide with the a-algebras generated by the vague

topologies in M and N respectively.

PROOF. The vague topology in M (or N) is by definition the product

topology induced by the mappings )J ~ pf f E F , so the class Cc

kEN, is a topological base. Since M (N) is

separable, every vaguely open set is contained in a(C) , and so a(C)

-42-

is the a-algebra generated by the topology. On the other hand we have

aCe) = M (or N) be Lemma 1.1. This completes the proof.

Unless otherwise stated, convergence in distribution of random measures

d (d d d . ~) . 1 h f han point processes enote ~,or sometlmes ~ wll ence ort mean

weak convergence of the corresponding distributions with respect to the

vague topology, i.e. ~n i ~ iff Ef(~n) ~ Ef(~) for every bounded and

vaguely continuous functional f: M~ R+ or N ~ R+ respectively. Con­

tinui ty in distribution holds for a large class of linear functionals.

Let Of denote the set of discontinuity points of the function f.

Let us further denote the mappings ~ ~ ~f and ~ ~ ~B by wf and

wB respectively.

~, ~l' ~2' ... be random measures on S such that

~ f i ~f for every bounded function f: S ~ R withn +

LEMMA 4.3. Let

d~ ~ ~. Thenn

compact support satisfying ~Df = 0 a.s. Furthermore~

kevery V E B~, kEN.

~ V i ~V forn

PROOF. Write C E B for the support of f and choose some open set

G E B with G ~ C By Urysohn's lemma, there exists some function

g E Fc such that Assuming v~ ~ ~ , we get for any boundedn

continuous function h E F

since gh E F , which proves thatcIf we further assume that

-43-

)JDf = a , it follows by Theorem 5.2 in Billingsley (1968), (which clearly

remains true for non-normalized measures), that

This proves that

(3)

The first assertion now follows by Theorem 5.1 in Billingsley (1968).

Specializing to indicators f = lB' B E B , it is seen from (3) that

B E B .

Hence the second assertion follows by applying Theorem 5.1 in Billingsley

k(1968) to the mapping )J + )JV, V E B~, kEN.

For point processes, convergence in distribution means the same thing

in M and N. In fact,

LEMMA 4.4. N is a vaguely closed subset of M. The class of point

processes on S 1.-S closed under convergence in distribution w.r.t. the

vague topology on M.

PROOF. Suppose that E N with v M By Lemma 4.3)Jl' )J2' )In + )J E

we have )J B + )JB for any B E B with )JaB = 0 , so )JB E Z for anyn +

such B Applying Dynkin's theorem as in the proof of Lemma 1. 2, it

follows that )JB E Z for all B E B , and so )J E N This proves that+

-44-

N is a vaguely closed subset of M. Now suppose that ~l' ~2' '"

are point processes on S converging in distribution to some random measure

~. By Theorem 2.1 in Billingsley (1968), we obtain

P{~ E N} ~ limsup P{~n E N} = 1 ,n-+oo

proving that ~ E N a.s., i.e. that ~ is a point process.

We shall now prove the basic fact that the classical notion of con-

vergence introduced above is essentially equivalent to convergence in

distribution with respect to the vague topologies.

THEOREM 4.5. Let ~, ~l' ~2' ... be random measures on S and Zet

I c B~ be a DC-semiring. Then the foZlowing statements are equivalent.

(i)

(ii)

d~ -+ ~ ,n

~ f i ~fn

f E Fc

(iii) L~ (f) -+ L~ (f)n

f E Fc

(iv) ~ V i ~Vn

kEN .

Two lemmas are needed for the proof.

LEMMA 4.6. Le t rl' r 2 , ... be random measures on S. Then {r}~ ~ . ~n

ia tight iff {t.: B}n

is tight for each B E B •

-45-

PROOF. We shall make use of the fact (cf. Bauer (1972)) that a set K

in M or N is relatively compact iff

(4) sup pB < 00

PEK

B E B .

Let us first suppose that {~n} is tight. By definition, there exists

for each £ > 0 some compact set K c M such that

(5) P{~ E K} ~ 1 - £n

n EN.

For fixed B E B we get by (4) m = sup pB < ':0 , and hence by (5)pEK

P{~ B > m} ~ P{~ 4 K} ~ £n n

n EN,

proving tightness of {~nB} .

Conversely, suppose that {~nBj is tight for each B E B. Let

Gl , G2

, ... be open B-sets with Gk + S , and choose for fixed £ > 0

the constants ml , m2 , ... > 0 such that

k , :n EN.

Define

K = n {p EM: p Gk ~ ~} ,k

and note that K is relatively compact, since there exists for any B E B

some k <:: N with Gk::J B , and therefore

sup llB ~ sup pGk ~ mk < 00 •

llEK llEK

-46~

Furthermore,

p{~ 4 K} = P u {~ Gk > ~ } S L P{~ Gk > mk}n k n K k n

proving tightness of {~n}

s; £ I 2-k= 2£

k

LEMMA 4.7. Let ~,n and ~l' ~2' be random measures on S, let

U c B be a DC-ring and let t > 0 be fixed.

and that ei the 1"

Further suppose that d~ 4- nn

(6)

or'

liminf P{~ U > £} S P{~U ~ £}n

U E U £ E (0,1) ,

(7) limsup E exp(-t~ U) ~ E exp(-t~U)J}+<lo n

U E U .

Then nF = 0 a.s. for every olosed Bet FEB with ~F = 0 a.s.

PROOF. Suppose that (6) holds and let FEB be closed with ~F = 0

a. s. By Lemma 4.1 there exists for every £ > 0 some closed set C E Bn

such that C ~ F and P{~C ~ £/2} s; £/2 We may next choose some U E U

such that U ~ C and P{~(U\C) ~ £/2} s; £/2. Then

P{~U ~ £} s P{~C ~ £/2} + P{~(U\C) ~ £/2} s £ .

Now ~ C ~ nC by Lemma 4.3, so we obtainn

P{nF > d s; P{nC > d s; liminf P{~ C > dnn4-<lO

s; liminf P{~ U > d S P{~U ~ d s: £ ,nn4-<lO

-47-

and since € was arbitrary, it follows that I1F = 0 a.s. as asserted.

A similar proof applies to the case when (7) holds.

PROOF OF THEOREM 4.5. The fact that (i) implies (ii) and (iv) was proved

in Lemma 4.3. Since (iii) follows trivially from (ii), it remains to

prove that (iii) and (iv) both imply (i).

Let us first suppose that (iii) holds. Then

L~ (tf) + L~(tf)n

f E Fc

so (ii) holds by the continuity theorem for L-transforms. But from (ii)

and Lemma 4.6 it is seen that { ~} is tight, and so it follows by Prohorov's"'n

theorem (Theorem 6.1 in Billingsley (1968)) that { ~} is relatively"'n

compact. This means that every sequence N' c N contains a subsequence

(ii)

By Theorem 3.1, this implies

E; .i E; (n E N) by Theoremn

was arbitrary, we getN'E; g n , and since

Nil such that ~ .in

we obtain E; f .i nfn

some n (n E: Nil) . From the implication (i) ==>

(n E Nil) , f € F , and since also ~ f ~ i;f ,c ndf E F ,we must have E;f = nf, f E Fc c

2.3 in Billingsley (1968).

Now this step is permitted by Lemma 4.3

To prove that (iv) implies (i), the same argument

for the point where we conclude from

k(n E Nil) , V E I , kEN.

d~ + n"'n

(n E Nil)

goes through, except

dthat E; V + nVn

provided I c B , and we shall show that this relation is indeed true.n

For this purpose, let U be the ring of finite unions of I-sets, and

note that these unions can always be taken disjoint. By (iv) and the

fact that addition is a continuous operation from to R , we ohtain+

-48-

d~ U ~ ~U, U c U , and son

liminf P{~ U > £} $ limsup P{~ U ~ £} $ P{~U ~ £}n nn-+oo n-+oo

U E U £ > a .

Hence it follows by Lemma 4.7 that nF = a a.s. for every closed set

FEB with E;F = 0 a.s., and in particular n31 = 0 a.s., I E I ,

as desired.

We shall now improve Theorem 4.5 in the particular cases of conver-

gence towards simple point processes and diffuse random measures. Given

any covering class C c B , and any a > 0 , we shall say that the sequence

{~n} of random measures on S is a-regular w.r.t. C if there exists for

every C E C some array {Cmk } c C of finite coverings of C (one for

each mEN) such that

(8) lim limsup L P{t; C k ~ a} = 0 .k

n mJII+<X> n-+oo

Note that, in the particular case when ~l = t;2 = ... = t; , this notion

reduces to a-regularity of E;, as defined in Section 2. As before, we

shall further say that {E;n} is regular, if it is a-regular for every

a > 0

THEOREM 4.8. Let t;, E;l' E;2' ... be point processes on S and suppose

that ~ is simple. Further suppose that U c BE; ~s a DC-ring and I c BE;

a DC-semiring. Then ~ ~ ~ iffn

(i)

(ii)

-49-

lim P{~ U = O} = P{~U = O}nn-+oo

limsup P{~ I > I} ~ P{~I > I}nn-+oo

U E U ,

lEI ,

(iii)

Moreover..

{~B} is tight for every B E B .n

t;, i ~ follows from (i) and (ii) if t;, '/-s 2-regular w.r. t.n

1 and from (i) alone if {~n} is 2-regular w.r.t. 1 or if

(9) 1imsup EE;, I ~ EO < 00nn-+oo

lEI .

PROOF. The necessity of (i) - (Hi) follows by Lenunas 4.3 and 4.6. Con-

versely, assuming (i) - (iii) • it follows by Lemma 4.6 that {~n} is

tight and hence relatively compact, so any sequence N' c N must contain

a sub-sequence N" such that ~ i some n (n E Nil) Here n is an

point process by Lemma 4.4. By (i) and Lemma 4. 7 we get naB = 0 a. s. ,

B E U u 1 , and so by Lemma 4.3

Hence

dt;, B -+ nBn

(n E Nil) BEUuI.

(10) r{nU O} = limnEN"

Pit;, U = O} =n

P{~U = O} U E U ,

p{ nI > I} = limnEN"

p{~ I > I} ~ limsup Pit;, I > I} ~n nn-+oo

P{t;,I > l} I E: 1 ,

and it follows by Theorem 3.3 that d~ = n Since N' was arbitrary,

this implies

-50-

Now suppose that ~ is 2-regular w.r.t. I and that (ii) holds.

Let I E I be fixed, and let {I.} c InJ

be an array of finite coverings

of I such that (2.8) holds with a = 2 and C . :: I .nJ nJ

If rm

is

the number of sets Imk for fixed mEN, we get by (ii)

limsup p{~ I > r }n mn+oo

~ limsup P U {~ I k > I}n+oo k n m

~ limsup L P{~ I k > I}n+oo k n m

Hence by (2.8)

lim limsup P{~ I > r } ~ lim L P{~I k > I} = 0 ,m-+oo n-+«> n m m-+oo k m

proving tightness of {~nI} . Therefore, (iii) follows from (ii) in this

case.

Next suppose that { ~} is 2-regular w.r.t."'n I and that (i) holds.

Then tightness of {~n} follows as above, so any sequence N' c N must

contain a sub-sequence Nil such that some n (n E Nil) As before

we may conclude from Lemma 4.7 that naB = 0 a.s., B E U u I so in

particular (10) remains true and ~ ~ n* follows by Theorem 3.3. Further-

more, it follows from (8) with a = 2 and Cmk :: Imk E I that

lim I P{nI k > I} = lim L lim P{, I k > I} ~ lim limsup L P{~ I k > I} = 0 ,m+"" k m m+= k nE Nil n m m-+oo n-+«> k n m

so n is 2-regular w.r.t. I, and hence simple by Theorem 2.4. Therefore

d~ = n , and again ~ ~, follows since N'

nwas arbitrary.

-51-

Finally suppose that (i) and (9) hold. By ~ebysev's inequality,

(9) implies tightness of {~ I}n

for every I E I , so is tight

by Lemma 4.6. Ifd

~ + n as n + 00 through some N" eN, then it followsn

as above that d * d~ = n and ~ I + nIn

(n EN") , I E I , so by (9) and

Fatou's lemma (in the formulation for convergence in distribution, see

Billingsley (1968), Theorem 5.3),

E~I*= En I ~ En! ~ liminf E~ I ~n

nEN"limsup E~ I ~ E~I

nn+OOI E I .

*lienee EnI = En I for all I E I , and n must be simple. As before,

we may conclude that The proof is now complete.

From Theorem 3.4 we obtain the following analogeous result which

may be proved in the same way.

THEOREM 4.9. Let ~, ~l' ~2' ... be point processes (random measures)

on S and suppose that ~ is simple (diffuse). Fu~ther suppose that

sand t are fixed real numbers with 0 < s < t and that U c B~ ~s

a DC-ring and C c B~ a covering class. Then _ d ~iff1- +

~n

-t~ U= Ee-t~U(i) lim Ee n U E U ,

n+oo-s~ C

> E -s~C(ii) liminf Ee n C E C- e ,n+oo

(iii) {~nC} is tight for every C E C .

Moreover, ~n 1. ~ follows from (i) alone if {~n} is 2-regular (Y'cgular')

w.r. t. C 01' if (9) holds with C in place of I

-52-

In many applications we need to consider convergence in distribution

with respect to the weak rather than the vague topology on M. When

wishing to emphasize the underlying topology, we write wd--+-

vdor --+- instead

of d-+ No new theory is needed for the case of the weak topology, since

this case may easily be reduced to that of the vague topology:

THEOREM 4.10. Let ~, ~1' ~2' ... be a.s. bounded random measures.

Then the following statements are equivalent.

(i) ~ ~ ~ ,n

(ii) ~ ~ and d~ ~ S -+ ~S •n n

(iii) ~ ~ ~ and inf 1imsup P{~ BC> d = a £ > an BEB nn-ro>

Note that (i) is trivially equivalent to

The point is that the latter condition may be replaced by the seemingly

much weaker conditions (ii) or (iii).

PROOF. Since ~ ~ ~ iff ~ ~ ~ and ~ S -+ ~S , the weak topology isn

metrizab1e, so the theory of convergence in distribution in metric spaces

applies to it. Write M(w) and M(v) for the space of bounded measures

in M endowed with the weak and vague topology respectively. Let us

first suppose that (i) holds. Since the mapping ~ -+ (~, ~S) from M(w)

into

(1968).

M(v) x R+

is continuous, (ii) follows by Theorem 5.1 in Billingsley

-53-

Next suppose that (ii) holds, and let £,0 > a be arbitrary. Since

{i; S} is tight, we may choose some a > a such thatn

(11) P{i; S > a} < 0n

n EN.

By dominated convergence, there further exists some B E B with

(12)_I;BC -a

E(l - e ) < £oe ,

and by Lemma 4.1 we may assume that l;aB = a a.s. Using (11), we obtain

a-I; S -1 -I; S-1 c n I; BC> I; S $ a] eaEI; BCe n

+ 0<' E[£ I; B e £ , + 8 $ £n n n n

I; BC -I; S -I; B -I; S~

-1 aE( n 1) e n+ 8 -1 aE( n en) 8E e e = £ e . e + ,

so by (ii), (12) and Lemma 4.3,

-I; B -I; SC -1 annlimsup P{I; B > £} $ £ e lim E(e - e ) + 8nn-+<x> n-+<x>

= £-leaE(e-I;B _ e-I;S) + 0 = £-leaE[e-I;B(l _ e-i;BC)] + 8

C$ £-l eaE(l _ e-I;B ) + 0 $ 8 + 8 = 28 .

Since 0 was arbitrary, this proves (iii).

Let us finally suppose that (iii) holds. Choose for each kEN

some Bk E BI; such that Bk t Sand

limsup P{l;n~ > k- 1} < k- 1

n-+<x>n EN.

-54-

Then

(13) lim limsup P{~nB~ > 8} = 0k-+-co n+co

£ > 0 .

We now define the metric p

by

in the topological product space M(v) x R+

lll' ll2 ( M

where is any metrization of M(v) Then

I~ Bk - ~ Sl = ~ BC

n n n k n,k EN,

so (13) may be written

Furthermore, we have in M(v) x R+

£ > 0 .

(15) (~, ~Bk) -+- (~, ~S) a.s. as k -+- co ,

and finally, since the mappings II -+- (ll, ll~) are continuous

M(v) -+- M(v) x R at all points II with lldBk = 0+

(16) (~n 'd

(E;, ~Bk) kEN .~nBk) -+- as n -+- co

Using Theorem 4.2 in Billingsley (1968), we may conclude from (14), (15)

and (16) that

(17)

Now the mapping

-55-

is continuous M(v) x R ~ M(w) , and therefore+

(i) follows from (17) by Theorem 5.1 in Billingsley (1968). This completes

the proof.

NOTES. The equivalence of (i) - (iii) in Theorem 4.5 was proved by Prohorov

(1961) for compact spaces and in general by von Waldenfels (1968), (see also

Harris (1971)). The remaining equivalence was given in Kallenberg (l973a).

This is also the source for Theorem 4.8, except for the assertion involving

(9) which was added by Kurtz (1973), while Theorem 4.9 improves and extends

a result by Kallenberg (1973e) and Grandel1 (1973). Theorem 4.10 is new,

although it was used implicitly in Ka1lenberg (1973c) and (1974b). We

finally refer to Jagers (1974) for Lemmas 4.2 and 4.4.

PROBLEMS.

4.1. Let P, PI' P2 , ... E M and let I E Bp be a DC-semiring. Prove

that p ~ p iff p I ~ pI, I E I .n n

4.2. Prove that p ~ p iff limsup p F :0; pF , FEB closed, andn nliminf 1J G ;:: :J.IG , G E B open. Note that one of these conditionsnis not enough.

4.3. Let F €: B be closed and G E B open, and let t E R Show+

that the p-sets {pF < t} and {pG > t} are open in M and N

(Hint: use Problem 4.2.)

4.4. Prove that, for bounded EM, w iff vp, PI' P2' ... Pn ~ P Pn ~ )J

and p S ~ pSn

on S and letwd

Ck~n -+ Ck~ ,for any bounded

W f = 0 a.s.

4.5.

-56-

Let ~'~l' ~2' ... be random measures

with Cko

t S. Prove that ~ ~ ~ iffvd n d

generally, ~ -+ ~ implies f~ ~ f~n n

f E F with bounded support satisfying

Cl , C2 , ... E B~

kEN. More

function

4.6. Prove that, if " '1' '2' are random elements in (S, B)

thend iff o ~ 0, + ,

n T ln

4.7. Prove Theorem 4.9.

4.8. Extend Theorem 4.8 to the case when ~1' ~2' ... are general random

measures.

d~n + ~ ,

a.s. if

4.9. Let ~'~1' ~2' ... be random measures on S such thatft .,.

and let 6: ~ 0 be fixed. Show that ~ = ~ - 0"'6:+ - "'(e:,oo) -

{~n} is a-regular w.r.t. B~ for every a > e: , and that the converse

is also true whenever E~ E M (Cf. Kallenberg (1975).)

4.10. Show that, if the class C in Theorem 4.9 is a DC-semiring, then

~ may be allowed to have atoms of any fixed size and position.

(Cf. Problem 3.9.)

4.11. Show that the equivalence of (i) and (iii) in Theorem 4.10 may essentially

be restated as a tightness criterion for random measures subject to the

weak topology in M. Give also a direct proof of this criterion.

4.12. Let ~1' ~2' .. . be Cox processes directed by n1 , n2 , ... respectively.d dShow that ~ + some ~ iff nn .+ some n , and that in this case ~ is

na Cox process directed by n . Prove the corresponding result for p-

thinnings with fixed p > 0

•

-57-

4.13. Let n, nl , n2, ... be random measures on S

suppose that the functions f, f l , f 2, '" € F

wi th uniformly bounded support and satisfy

dwith ~ + nand

are uniformly bounded

(Cf. the proof of TheoremShow that in this case

n{s € S: fn(sn) + f(s) for some sl' 52' ... + s} = 0 a.s.

d~fn + nf .

8.1 below.)

5. EXISTENCE

In the limit theorems of the preceding section, the limiting random

measure ~ was always assumed to exist. We shall now show that, under

mild addi tional conditions, the existence of E; will hold automatically

hy compactness. This provides a new approach to the general existence

problems in random measure theory.

We begin with the counterpart to Theorem 4.5.

LEMMA 5.1. Let E;l' E;2' ••• be random measures on S and let U c B

be a DC-ring. Suppose that either

(i) f € Fc or

(ii)d

~nV + some SV'

c,v are such that

un: th uk.y au .

k € N , where the random vectors

whenevey;' U, Ul ' u2 ' ..• € U

Then l; §. somen

l; satisfying

-58-

f E Fc 01'

kV E U ,

kEN , respectiveLy.

PROOF. In both cases, {l;n} is tight by Lemma 4.6, so by Prohorov's

theorem, /; §. some l; as n -+ 00 through some sequence Nt c N Butn

then /; f §. /;f (n E Nt) , f E F , by Theorem 4.5, so assuming (i).n c

we obtain I;f ~ I;f , and the assertion follows by Theorem 4.5.

In case of (ii), we may proceed as in the proof of Lemma 4.7 to conclude

that 1;3U = a a.s., U E U But then I; V ~ I;V (n E Nt) , V E Uk ,n

kEN, by Theorem 4.5, and the assertion follows as before.

Arguing in the same way, we can easily derive analogeous results

for convergence towards simple point processes and diffuse random measures.

For example,

LEMMA 5.2. Let 1;1' I;Z' be point processes on S, and Zet U c B

be a DC-ring and I c B a DC-semiring. Suppose that {I;n} is Z-reguLar

1.1'. r. t. I and that

(1) P{I; U = O} -+ some ~(U)nU E U .

Further suppose that ~(Un) -+ 1 whenever VI' Uz, ... E U and B E U u I

with U -t aBn

Then there exists some simpZe point process l; on S

(2)

dI; -+ I; andn

P{l;V = a} = ~(U) U E U .

-59-

We shall now use Lemma 5.1 to derive two important special cases

of a general existence theorem corresponding to the second part of Theorem

3.1. For notational convenience, we consider random vectors in 8S

random measures on the space {I, ... ,k} , kEN

THEOREM 5.3. Let U c B be a DC-T'ing and foT' any V E Uk, kEN,

let ~V be a T'andom vectoT' in R~ Then theT'e exists some T'andom meaSUT'e

~ on S satisfying U c B~ andkV E U , kEN, iff

(i) the distT'ibutions p~-l aT'e consistent,V

a . s . wheneve:'('(ii)

(iii)

~V{l} + ~v{2} = ~v{3}

with disjoint VI and V2 ,

~V ~ 0 wheneveT' V, VI' V2, ... E U withk

V -I- av .n

PROOF. The necessity of (i) - (iii) is obvious. Conversely, suppose

that (i) - (iii) hold. For fixed V E U , let

of nested partitions of V , choose arbitrary

{U .} c UnJ

Snj E Vnj

be a null-array

for all nand

j , and define the random measures ~l' ~2' ... on S by

~n = ~ ~v{j}os . ' where V = (UnI , Vn2 ' ... ) , n EN.J nJ

Writing U for the ring generated by V I' U 2' ... , we obtain by (i)n n n.

and (ii)

kEN ,

-60-

and so

d~ V + ~n V kEN ,

where U' = U U Now U' is a DC-ring in the subspace V , so itnn

follows by (iii) and Lemma 5.1 that ~ ~ some ~' satisfyingn

(3) kEN .

To see that (3) extends to U n V , let mEN and VI' ... ,Um E U n V

be arbitrary, and repeat the above argument with a refined null-array

such that U' is replaced by some U":::> U, U {Ul'

the existence of some random measure ~" such that

,V } . This yieldsm

(4) ~"V 9: t;<, - V kEN .

Comparing (3) and (4), we obtain

t; 'V d t;"V kEN ,

and so t;' ~ E;" by Theorem 3.1. Inserting this into (4) yields the desired

extension of (3).

To complete the proof, let VI' U2 , ... E U with U~ + S , and let

t;i, t;2 .. , be random measures t;' obtained as above with VI' U2, ...

as lJ. Then it follows by another application of Lemma 5.1 that t;k ~ some t,

with the asserted properties.

Rk . Then there exists some random measure+

V E Bk

, kEN, iff

THEOREM 5.4. For anyk

V E B ,

-61-

kEN , let ~ be a random veator in

d·on S satisfying ~V = ~V '

(i) the distributions PE;~l are aonsist;ent,

(i i) a. 8. whenever'

with disjoint B1 and B2 ,

(iii)d

E;B + ° whenevern

B1

, B2

, ... E B with B '" ~ .n

PROOF. The necessity is again obvious. Conversely, suppose that (i)

- (iii) hold. Define U = {B E B: E;aB = 0 a.s.} , and verify that U

is a ring. To see that U has the DC-property, let t E S be fixed

and put S(t,r) = {s E S: p(s,t) < r}

and suppose that 8 > ° is such that

Choose £ > 0 with S(t,£) E B ,

(5) P{E;aS(t,r) > 8} > 6

for infinitely many r E (0,£) , say for r l , r 2 , ... By (i) and Kolmogorov's

consistency theorem, there exist some random variables E;O' ~l' E;2' .••

such that

and we get by Fatou's lemma

-62-

00

contradicting the fact that, by (ii), L ~n ~ ~O < 00 a.s. Hence (5)1

can hold with fixed 0 > a for at most finitely many r E (O,E) , and

so the set {r Eo: (O,E): ~S(t,r) = a a.s.} must be dense in (O,E)

This proves that U contains a base and hence has the DC-property.

Applying Theorem 5.3, it is seen that there exists some random measure

~ on S satisfying

(6) kEN .

To extend (6) to B, let k E Z+ and V E Uk be fixed and let

B, Bl , B2, '" E B with

we get

B t B .n

By Theorem 4.4 in Billingsley (1968)

and so by continuity

(7)

d~ \ + ~ ~,V,B,B B V,B,pn

d~V B + ~V B .

, n '

A similar argument shows that (7) is also true when

other hand

B oj. B .n

On the

~(V, B ) + ~(V,B)na. s. , B + B ,

n

so by the monotone class theorem in Lo~ve (1963) we have dt;(V,B) = ~V B ',

B E B Proceeding inductively, we obtain the desired extension of (6).

-63-

Turning to the existence problem for point processes, let us first

consider a necessary condition. Define for any set function ~ on B

llA~(B) = ~(A u B) - ~(B) A,B E B •

Similarly, let llA ... llA ~(B) be the set function obtained by forming1 n

successive differences with respect to B. Say that ~ is corrpletely

monotone if

n EN.

LEMMA 5.5. Let ~ be a random measure on S and let ~ E (0,00] be

fixed. Then the set function -t~BEe , B E B , is completely monotone.

PROOF. Let us first suppose that t = 00

we get

For the first order difference

o s P{~A > 0, ~B = O} = P{~B = O}

= P{~B = O}

P{~A = 0, ~B = O} =

P{~(A u B) = O} = -l:IAP{~B = O} ,

and continuing inductively, we obtain

o s P{ ~A1 > 0, .,. ,~n > 0, ~ Bn= O} = (-1) l:I ... llA P{~B = O}

Al nn EN.

In the case t < 00 ,let n be a Cox process directed by t~ and note

that

p{nB = o}= Ee-t~B B c B .

-64-

Now suppose conversely that ~ is a completely monotone set function

defined on the DC-ring U and such that ~(Un) + 1 whenever U, Ul , U2' ... E U

with U + au. Let V E U be arbitrary and letn

be a null-array of nested disjoint partitions of

Unl ' ... ,Unkn

V. For fixed n EN, choose

process on S with all its mass confined to snl'

r{1; {s .} = 1, j E J; ~ {s .} = 0, j ¢ J} =n nJ n nJ

non-random points

(8)

s . E U . ,nJ nJ

j = 1, ... ,k .nLet ~n be a simple point

, snk and such thatn

= P{E; U . > 0, j E J; 1; U U . = O} =n nJ n j¢J nJ

= {j~J [-OUnj]}~[j~J Unj]for each subset J c {I, ... ,kn } . Note that ~ exists, since the<"n

right-hand sides of (8) are non-negative, and summation over J yields

1. (The latter fact is easily established by induction in the number

of partitioning sets.) By (8),

U E Un

n EN,

where

with

Un is the ring generated by Unl ' ... ,Unk ' so (1) is satisfiedn

U replaced by U, = U U If {1;} is known to be tight, it followsn nn

as in the proof of Theorem 5.3 that (2) is satisfied for some point process

*1; on S. Since (2) remains true for 1; , we reach the following result.

THEOREM 5.6. Let ~ be a set function on some DC-ring U c B. Then

there exists some simple point process E; on S satisfying (2) and such

that

-65-

U c B iffS

(i) ~ is completely monotone on U 3

(ii) ~(Un) ~ 1 whenever U, Ul , U2' ... E U with Un ~ au ,

(iii) for every V E U there exists some null-array {Unj } C U

of nested disjoint partitions of V such that the sequence

of point processes sn defined by (8) is tight.

NOTES. Theorem 5.4 is due with an entirely different proof to Harris

(1968) and (1971). For point processes, a much more general result (con­

taining in particular the point process case of Theorem 5.3) is given by

Kerstan. Matthes and Mecke (1974), page 17. See also Prohorov (1961)

and Mecke (1972) for further general existence criteria. Finally, Theorem

5.6 is contained in the general results by Kurtz (1973) and Karbe (1973),

(cf. Kerstan, Matthes and Mecke (1974), page 35).

Our approach via Lemmas 5.1 and 5.2 seems to be new, although the

idea to prove existence by means of tightness arguments is well-established

in other fields, (see e.g. Billingsley (1968)).

PROBLEMS.

5.1. Prove Lemma 5.2 and the corresponding result for diffuse random

measures.

5.2. Show that, in the particular case of non-random t;v' Theorem 5.3

reduces to a version of Caratheodory's extension theorem.

~66-

5.3. Prove directly from (i) and (ii) of Theorem 5.4 that (ii) holds

with any finite number of terms. By (i) and Kolmogorovts consistency

theorem, PI;;-l can be consistently defined even for V E Ba> ShowV

by means of (iii) that (ii) then extends to infinite sums.

5.4. Suppose that S is discrete, i.e. that all its points are isolated,

and let ~ be a set function on U = B Show that (2) holds for

some point process I;; on S iff <jl is completely monotone and

<PC"') = 1 .

5.5. Show that Theorem 5.6 is false in general without condition (ii).

(Hint: let S = R and place a fictitious atom at the point 0+.

Let U be the ring generated by all intervals (a,b].)

5.6. Show that Theorem 5.6 is also false in general without condition

(iii) . (Hint: let I;; be a stationary "point process" on the unit

circle, whose atom positions have exactly one limit point.)

5.7. Show by a direct argument that the set function

is completely monotone for any random measure I;;

-tl;;BEe ,B E B ,

and fixed t < a>

6. NULL-ARRAYS AND INFINITE DIVISIBILITY

In this section we consider the general central limit problem for

random measures, i.e. the problem of convergence in distribution of sums

L I;; . , where the I;;nJ· , j,n £ N • form a nuZZ-appay of pandom measupes. nJJin the sense that the I;;nj are independent for fixed n and such that

(1) lim sup P{I;; .B > E} = 0n~ j nJ

E > 0 B E B .

-67-

Note that, in the particular case of point processes, (1) reduces to

lim sup P{~ .B > O} = 0n-+<>o j nJ

B E B .

Just as for distributions on R, the class of limiting distributions

of the sums \ ~ coincides with the class of infinitely divisible dis-Lj nj

tributions. A random measure ~ is said to be infinitely divisible4 if,

for every fixed n EN, we have for some independent

and identically distributed random measures ~1' In the point

process case, we require ~1' ... '~n to be point processes. Note that,

since a point process may be regarded as an N-va1ued random measure, we

have two notions of infinite divisibility for point processes, (which

are not equivalent since the measure n~ may belong to N even for ~

in M\N). If nothing else is stated, we mean infinite divisibility as

a point process.

In our derivation of canonical representations and convergence criteria,

we shall proceed in steps, beginning with the simple particular case of

R -valued random variables. In this case, (1) takes the form+

£ > 0 .lim sup p{~ . > £} = 0n-+<>o j nJ

We shall use K to denote the function 1 - e-x Furthermore, the pro-

jections and

= x·· t

are defined by

4 For a different notion of infinite divisibility, see O. Kallenberg,Infinitely divisible processes with interchangeable increments andrandom measures under convolution, Tech. Report, Dept. of Mathematics,Gtlteborg, 1974.

-68-

LEMMA 6.1. An R+-valued random variable ~ is infinitely divisible iff

(2)-'IT

- log Ee-~t = ta + A(l _ e t) t ~ 0 ,

for some a ~ 0 and some A E M(R')+

with AK < ~ , and in that case

a and A are unique. Moreover, any a and A with the stated properties

may occur in (2). If

then L.~ . &some ~J nJ

{~ .}nJ

iff

is a null-array of R+-valued random variables,

(3) \' -1 wK L'P~ . -+ some

J nJ

and then ~ , a and A satisfy (2). In the case of Z -valued random+

variables, (2) holds with a = 0 and bounded A E M(N) , while (3) is

equivalent L -1 w Nto . p~ . -+ A onJ nJ

Note that (3) is equivalent to the conditions

(i) L'P~-~ ~ A on R' with A{~} = 0 ,J nJ +

(ii) lim limsi L E ~ . = ae: nJ

,e:+O n~ j

where R' = (0 ,~J , limsi stands for both limsup and liminf , and+

E denotes the truncated expectation at the level e: > 0E:

PROOF. Writing ~ . = L~ ,we claim thatnJ ",.nJ

d\' ~ -+ somef "'njJ

iff

(4 ) ~n =? (1 - ~nj) -+ some continuous ~ ,J

and that in this case ~ = L~ = e-~ . In fact, \' ~ ~ ~ impliesLj nj

-69-

II .4> • ....." , whenceJ nJ

- I log ~ ..... - log 4> = ~ ,j nJ

and here ~ is continuous since 4> is an L-transform. Since (1) is

4> , and it follows that

(4) follows from this by con-4> .) .... a ,nJ

log 4> .nJ

Conversely, by a similar argument,

LI; . & some I;. nJJ

R and note+

~n = ~ p~-l. n E·N on1\ L "', ,

j nJ

(4) implies IT 4> ..... some continuousj nJ

We next introduce the measures

clearly equivalent to max(l -j

sidering a Taylor expansion of

that for t ~ 0 ,

<Xl

-tl; . -tl; .?J(1 _ e-tX)pl;-:(dx)~ (t) = L (1 - Ee nJ ) = LE (1 '" e nJ) = =n

j j nJJ 0

<Xl <Xl

f (1 - e-tX)~ (dx) J-txl-e

(K~n) (dx)= = .n l-e -x0 0

Hence (3) implies (4) with 1/1 given by<Xl <Xl

f -txf ( -tX)~(t)

l-e(d) (dx) o ,= ta + = ta + '1 - e . ~ (dx) t ~

-xo l-e 0

~ being continuous by monotone convergence. Conversely, suppose that

(4) holds for some continuous ~, and note that

~ (t + 1) - ~ (t)n n= f (e - tx - e- (t+1) X) ~n (dx)

o= J

o

-txe (KA) (dx)

If ~(l) ~ ~(O) , we may apply the continuity theorem for L-transforms

to the probability measures K~ /~ K , and conclude that KAn convergesn n

weakly on R+ But this is also true when ~(l) = ~(O) , since then

-70-

The uniqueness of a and A in (2) follows from the formula00

1/J(t + 1) - 1/J(t) = fo

-txe (a8

0+ KA)(dx)

and the uniqueness theorem for L-transforms. To see that every choice

of a and is possible, use (3) with -1An = [n , oo)A , n EN, to

construct a ~ with a = 0 and arbitrary A , and then replace ~ by

~ + a. In particular, the infinite divisibility of ~ in (2) follows

by dividing this relation by an arbitrary n E N Conversely, any infinitely

divisible L-transform ¢ = L~ can be difinition be written in the form

¢ = ¢n for every n E Nn so the above argument applies with ¢nj = ¢n

j = 1, ... ,n , showing that ~ has indeed the representation (2). The

last assertion follows easily from (3).

We state explicitly the corresponding tightness criterion.

LEMMA 6.2. Let {~nj} be a nuZZ-array of R+-vaZued random variabZes.

Then {I.~.} is tight iffJ nJ

(i) limsup L Et~ . < 00

n-+OO j nJt > 0 ,

(ii) lim limsup L P{~ . > t} = 0 .. nJ

t -+00 n-+OO J

If the ~nj are Z+-vaZued, then (i) may be repZaced by the condition

(i)' limsup I P{~ . > O} < 00 •

11+00 j nJ

PROOF. The sequence {d }n

-71-

of the last proof is vaguely relatively compact

on R+ iff (KAn) rO,t] is bounded for each fixed t > 0 , i.e. iff (i)

holds. The role of (ii) is to prevent mass from escaping to +~, thus

ensuring every vaguely convergent sub-sequence to be weakly convergent.

For point processes, (i) and (ii) imply (i)', which in turn implies (i),

so in this case, (i)' may replace (i) .

We now turn to the m-dimensiona1 version of Lemma 6.1, mEN

Let ~ be a one-point compactification of Rm and for+ '

t Rmx, E+

write

xt = x • t for the inner product of x and t

(2) holds for Bome

LEMMA 6.3. mAn R+-valued random vector

ma E R+ and some

~ is infinitely divisible iff

m -1A E M(R+\{Ol) with (A~t)K < ~

t E R: ' and in that case a and A are unique. Moreover, any a and

A with the stated properties may occur in (2). If {~njl is a null­

array of R:-valued random vectors, then Lj~nj ~ some ~ iff there exist

some a and A as above such that

(i) L·P~-~XA on ~\{O} ,J nJ

(ii) lim limsi LE ~ .{kl =ark} k = l, ... ,m£+0 Jl+<X>

. £ nJJ

and then and satisfy (2) • In the case ofm randomE;, , a A Z -valued+

vectors, (2) holds with a = 0 and bounded A E M(Zm\{Ol) , and we may+

replace (i) and (ii) by the condition L'P~-~ ~ A on Zm\{O} .J nJ +

-72-

PROOF. Let {~ .} be a null-array satisfying (i) and (ii) for somenJ

a and A with the stated properties. Then it is easily seen that

so by Lemma 6.1,

continuity theorem for m-dimensiona1 L-transformsand it follows hy the

dthat I·~ . ~ some ~

J nJ

By Lemma 6.2, there

-TI

E exp(- I t~ .) ~ exp(-ta - A(l - e t)). nJJ

satisfying (2).

dConversely, suppose that I.~ . ~ some ~.

J nJ

exist for every sequence N' c N some subsequence Nil C N' and some

a and A the stated properties such that (i) and (ii) hold as n ~ ~

through Nil. By the direct assertion of the present lemma, it follows

that and satisfy (2). Now at is unique for every

by Lemma 6.1, and therefore a must be unique. As for A, write

1 := (l, ... ,1) E Rm and note that by (2)+

Applying the uniqueness theorem for m-dimensional L-transforms, it follows

that the bounded measure-TI

since 1 - e 1:.. > 0 on-

pendent of the choice of

-'IT 1(1 - e -)A is unique, and hence so is A

R:\{O} . This proves that a and A are inde-

N' , and hence that (i) and (ii) remain true as

n + 00 through N, (cf. Theorem 2.3 in Billingsley (1968)). The remaining

assertions now follow as in the proof of Lemma 6.1.

-73-

Before proceeding to arbitrary random measures, we prove a lemma.

LEMMA 6.4. Let ~ be a random measure on S and let I c B be a DC-

semiring. Then is infinitely divisible iff ~V is for evepy

kEN . In the point process case it suffices that be infinitely

divisible for every II' ,Ik E I, kEN.

PROOF. The necessity of our conditions is obvious. Conversely, suppose

that ~V is infinitely divisible in for every kV E I , kEN .

Since infinite divisibility of random vectors is preserved under convergence

in distribution, (as may be seen from the continuity theorem for L-transforms),

we may use the monotone class theorem in Lo~ve (1963) to extend this property

to arbitraryk

V E B , kEN It follows that there exist for every

n E N some random vectors ~V n satisfying,

(5)-n

(L~V) = L~V,n

k,n EN.

From (5) it is easily seen that the family {~V n} satisfies the conditions,in Theorem 5.4, and hence there exists some random measure ~n on S with

~ V ~ k k E N Writing (5) in the form~V n ' V E B ,n ,

n V E Bk k,n EN,L~V :: (L~ V)n

and using Theorem 3.1, it is seen that ~ is infinitely divisible.

Turning to the point process case, suppose that I·o.J J

is infinitely

divisible (as a random variable in Z+) for every II'

kEN. Since repetitions of I-sets may occur, this proves that, for

-74-

any fixed V = (11' ... ,Ik) ( Ik , kEN , tW = L·t.f;!. is infinitelyJ J J

divisible in Z+ for every t = (t l , ... ,tk) € Nk Turning to rationals

and using the above closure property, it follows that tt;V is indeed

infinitely divisible in R+ for every t = (t l , ... ,tk) € R~. Hence,

by Lemma 6.1, there exist some constants

such that

A ;:: 0p

- log Ee-utt;V = L (1 _ e-utp)Ap p

Moreover, making

In fact, there can be no constant component a here since the

infinitely divisible in Z+ and hence P{t;I. = O} > 0 .J

t;I.J

are

a Taylor expansion in (2), it is easily seen that the support of A is

contained in the support of pt;-l. (This is a simple special case of

Theorem 6.9 below.) Now suppose that n is an infinitely divisible random

vector in Z: with spectral measure A = \ A 0 , the existence of whichL.p P p

is ensured by Lemma 6.3 since LpAp< 00 Then Ltn coincides with

Ltt;V above, and so tn ~ tt;V . If the t. are chosen rationally independent,J

it follows that d t;V which proves that t;v is infinitely divisible.n :: ,

Since V was arbitrary, we may now proceed as in the first part of the

proof to conclude that ~ itself is infinitely divisible.

We are now ready to extend Lemmas 6.1 and 6.3 to random measures

defined on general spaces S.

THEOREM 6.5. fi pandom measure ~ on S is infinitely divisible iff

(6)-H

Log Ee '-1r

= af + A(l _ c f) f E F ,

-75-

-'

for some a E M and some measure _A on M\ {o} satisfying

B E B , and in this case a and A are unique. Moreover~ any a and A

with the stated properties may occur in (6). If

ofY'andom measures on S ~then I.F;. . ~ some t;,J nJ

a and A as above such that

{E;,nj} bS a null-array

iff there exist some

(7) \ -1 w -1KL'P(t;, .f) ~ afo O + K(A~f )J nJ

f E Fc

and then t;, ~ a and A satisfy (6). ~f I c B is a DC-semiring satiSfying

aaI = A{~aI > O} = 0 1 E I , then (7) is equivalent to the conditions

(8) I -1 v -1 ~\{o} y E rk kEN ,. P(t;, .Y) ~ A~V onJ nJ +

(9) lim limsi L E t;, .1 = al 1 E IE:~ n-+<>o j E: nJ

In the point process case~ we have a = 0 in (6) while A is confined

(9) can be replaced by (8) alone~ provided the vague convergence on

is strengthened to weak convergence on z:\{O}

to N\{O} and satiSfies -1(A~B )N < 00 B E B. In this case~ (8) and

It\{O}+

PROOF. Suppose that t;, is an infinitely divisible random measure. Then

E,Y is infini tely divisible for every Y = (Bl , ... , Bk) E Bk

, kEN

so by Lemma 6.4 there exist some unique ~ E R: and Ay E M(R:\{O})

such that

(10)-~

1 E -t(t;,V) t ~ (1 t)- og e = ay + A y - e

-76-

Choosing all components t{j} but one equal to 0, we obtain

ay = (aB ' ,aB ) , while choosing exactly one component equal to1 k

a , it is seen that {Ay } is in the obvious sense a consistent family

of measures. Let us now choose Y = (A,B, A u B) in (10), where A

and B are arbitrary disjoint sets in B. Defining by

we get for any t

A'C = AA B{(x,y): (x,y, x + y) E C} ,,

= (u v w) E R3

, , +

-7TA--Cl t) -__ log Ee-t(~V) -_uaA + vaB + waAuB + .-V - e

= _ log Ee-Cu+w)~A-(v+w)~B

= (u + w)aA

+ (v + w)aB

+ J (1 - e-Cu+w)x-(V+W)YPA,BCdXdy)

-7T

= uaA + vaB + wCaA

+ aB) + A'(l - e t)

By the uniqueness assertion in Lemma 6.4, it follows that

(11)

and A = A' and in particularV '

(12)

= A'{(X,y,z) E R3 : x + y ~ z} =+

= AA B{(x,y) E R:: x + y ~ x + y} = 0 .,

Next suppose that Bl , B2, ... E B with B (~ ~ , and conclude from (10)n

that -t a while

(13)

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E: > 0 .

Combining the former relation with (11) , it is seen that the set function

B -+ Cl B , B E B is in fact a measure. As for the family fAv} let

B f B be fixed and define the measures A' E M(Rk) , V E Bk , k E N , hyV +

Since carries over to

{AV} , we may apply Theorem 5.4 to the latter family, and conclude from

(12) and (13) that there exists some unique measure A' onB

{IJB > O} E M

satisfying Defining the mea~ure on the same set by

A (d) (1 - e- IJB) -lA' (d1l)B IJ = B ~

we ohtain for any measurable set

f (1{IJVEC} J

-x -1= (1 - e ) An v(dX x C) =

R' ,+

-x -1 -y(1 - e ) (1 - e )AB B V(dxdy x C) =, ,

-- A (R' 2x C) = A (R' x C)B,B,V + B,V + '

and it follows that for any B' E B with B' ~ B

This proves that the measures AB are all restrictions of A = sup 1\B .BEB

-78-

It is now easy to verify that this A is the unique measure satisfying

(6). Using Theorem 5.4, it is further seen that any a and A with

the stated properties define the distribution of some random measure

~ satisfying (6), and it follows in particular that any such ~ is infinitely

divisible.

measure satisfying (6), it

and hence by Theorem 4.5,

L.~ . ~ some ~. ThenJ nJ

Now suppose that

is infinitely divisible by Lemma 6.4, and

follows by Lemma 6.1 that

d2.~ . ~~. Conversely, suppose that

J nJ

{~nj} is a null-array of random measures satisfying

A with the stated·properties. If ~ is a random

L'~ .f ~ ~f, f E FJ nJ c

and(7) for some

hence it satisfies (6) for some a and A. But then (7) follows by

Lemma 6.1. The same argument may be used to show that (8) and (9) are

necessary and sufficient, and therefore are equivalent to (7). Finally,

the assertions for point processes may easily be deduced from (8).

We shall now give two useful and illuminating interpretations of

the canonical representation (6). Let us write lCa,A) for any infinitely

divisible random measure ~ satisfying (6). In the point process case we

write 1(0,1..) = 1(1..) .

LEMMA 6.6. Let a and A be such as in Theopem 6.5 and let n be a

Poi8son process on M\{O} with intensity A. Then

(14) l(a,A) ~ a + f lJn(dlJ) .

PROOF.

-79-

~

Denote the 1eft- and right-hand sides of (14) by ~ and ~

Since for fixed f € F the mapping 1Tf : ].l -+ ].lf is measurable, it follows~

as in Section 1 that the function ~f = O'.f + n1Tf is a possibly infinite-

valued random variable, and we obtain by (1.2) and (6)

-~f -af -mTf -af ( -1rf ) -~f(15) Ee = e Ee = e exp -A(l - e ) = Ee .

In particular~ d~B = ~B < 00 a. s. , B E B , and it follows as in Lemma

~

1.3 that ~ is a random measure. Finally, (14) follows from (15) by

Theorem 3.1.

LEMMA 6.7. Let 0'. and A be such as in Theorem 6.5, and let Ml , M2, ...

be a disjoint partition of M\{O} such that AM < 00 , n EN.n For each

n ( N , we further assume that vn is a Poissonian random variable with

mean AM and thatn

common distribut'ion

dependent, we have

M VAM •n n

are random measures on Mh with the

Letting all these random elements be in-

(16) I(a,A) d a +

PROOF. The right-hand side of (16) has the L-transform, evaluated at

an arbitrary f E F ,

-80-

00_afe II

n=l

-AM 00

e n ~

k=O

00

e-a.f IIn=l

00

-af= e II

n=lexp{ - (M A) (1

n

-af= e exp{ -A (1-'IT

exp{-a.f - A(l _ e f)}

and by Theorem 6.5, this agrees with the L-transform of 1(a,A) . Thus

the assertion follows by Lemma 1.3 and Theorem 3.1.

The last two lemmas may be used to establish various relationships

between an infinitely divisible random measure ~ and the corresponding

canonical measures a and A. We shall give two results of this type.

~ g *THEOREM 6.8. Let 1(et,A) on S and let a > 0 Then ~a = 0

* * "A{ll{S} > o}a.s. iff (i) a = o ., (ii) A{l-! ;t O} = 0 ., and (iii) = o .,a a

s E S . In pa:I'ticular., ~ E Md a.s. iff a E Md and A~ = 0 •

*PROOF. Suppose that ~a = 0 a.s. By Lemma 6.7 we obtain (i), and

furthermore, *( ~.) = 0 a.s. whenever AM > O. But thensnJ a n

M A{l-! ;t O} = 0, n EN, and summing over n yields (ii). To proven a

(iii), suppose conversely that ~A{l-!{S} > E} > 0 for some n EN,

s E Sand E > O. Then we get the contradiction

P{~: ;t O} ~ p{~{S} ~ a} ~ p{Vn > [a/E] , ~nj{s} > E , j S[alE] +l} =

• P!Vn > [alE)}! P{Enj!S} > Ellla/EJ.1 > 0 •

-81-

so (iii) must indeed be true.

Conversely, suppose that (i) - (iii) hold. From (ii) it follows

that a.s. whenever AM > 0 , so it suffices to prove that,n

wi th probability one, no two terms in (16) can have atom positions in

common. But this follows by Fubini's theorem from the fact, implied by

(iii), that ~ .C = 0 a.s. for any countable set C c S .nJ

To state the next result, let SeA) denote the support of a measure

A on M w.r.t. the vague topology in M. Moreover, given any MeM,

define GM as the additive semigroup in M generated by M u {a} , and

write a = min M whenever a E M and a ~ ~, ~ EM.

THEOREM 6.9. Let dt; = l(a,A) on S. Then

(17)

(18) GS(A)-1= S (P t; ) - Ct •

PROOF. To avoid trivialities, we may assume that A ~ O. Let us first

consider the case of bounded A By Lemma 6.7, we may then assume that

t;. are independentJ

A/AM. From (19) it

d v(19) ~ a + L t;. ,

j=l J

where v is Poissonian with mean AM while the

of v and mutually independent with distribution

-AMis seen that ~ ~ a a.s. and P{t; = a} = e > 0 , proving (17). Since

addition and subtraction of Ct are continuous operations on M and

~82-

M + a respectively, we further obtain - a

and so it suffices to prove (18) in the case a = 0

Suppose that ~O E GS(A)\{O} , say ~O = ~l + ••. + ~k ,where kEN

and ~l' ... '~k E SeA) By the continuity of additien in M, there

Since Sep~-l) is closed, this proves that

exists for every E > 0 some <5 > 0 such that

P{p(E,;,\10

) < d ~ P{v = k , P(~j , ~. ) < 0 , j = 1, ... ,k} =J

k= p{v = k} II P{p (~ . , ~. ) < o} =

j=l J J

-AM (AM)k k= e k! II A{p ell, \1') < o} / AM > 0 ,

j=l J

proving that ~O E S(p~-l)

GS(A) c S(p~-l) .

Conversely, suppose that 110 E S(p~~l) . For n EN, let

obtained as ~ in (19) but with v replaced by v An. Then

a.s., so we get

(20) o < P{p(~,~O) < £} ~ liminf P{p(nn' llo) < £}n~

£ > 0 ,

and hence there exist some n EN, with If

we can show that lln E GS (A) , n € N it will follow that \1 0 E GS(A) ,

proving the relation S(p~-l) c GS(A) Keeping n E N fixed, we thus

assume that-1

Then there exists form E S(Pn ) every pEN somen

k E {O,l, ... ,n} satisfying

(21)

-83-

-1 .o < P{p(~, m) < p , \I = k} t:: P{P(~l + ... +-1

~k' ID) < P , \I = k}

-1= P{P(~l + ••• + ~k' m) < p }P{\I = k} ,

and since the set of possible k-va1ues is non-increasing in p, we can

choose k independent of p. Moreover, k = 0 implies m = 0 E GS(A) ,

so we can further assume that k ~ o. Let us now define the sets

Since

k -1Ap = {( 11 1 , ... , llk) EM: p (11 1 + ••• + llk' m) < p }

is separable in the product topology, each Ap

by countab1y many sets

pEN .

can be covered

j = I, ... ,k}

with (m1, ... ,mk) E Ap , 80 by (21) there must exist some vector

,mk ) E A satisfyingp p

(22) -1o < P{p(~, m. ) < pJP

kj = I, ... ,k} = II

j=l

-1P{p(~., m. ) < P }.

J JP

From vwe get m1p + ... + mkp ~ m as p ~ 00 , so

1imsup m. f ~ lim (m1 + ••• + mkp)f =~oo JP ~ P

mf < 00 f E Fc j = 1, ... ,k ,

showing that the sequences {mlp}' ,{mkp } are relatively compact. We

may therefore choose some sequence N' c N and some mI , ... ,mk E M

such that v (p EN') j 1, ,k It follows in particularm. ~ m. , =JP J

that v (p E N' )m1p + ... + mkp ~ m1 + .. . + mk , so m = ID1 + ... + mk

Let us now fix j E U, ... ,k} and E: > 0 , and choose pEN' so large

-84-

that p (m. , JD.) < £/2JP J

and -1P < £/2 . By (22), we then obtain

-1P{p(l;., m.) < d ~ P{p(~., m.) < E:/2} ~ P{p(~., m.) < p } > 0J J J JP J JP

so we get This shows that m E: GS(A) and hence

completes the proof in the case of bounded A .

In the case of unbounded A we still have ~ ~ a a.s. by Lemma

6.7, and if (18) is true, we get in addition -1a E: S(P~ ) , proving (17).

It thus remains to prove (18), and again we may assume that a = O. For

n E: N .~kj ,

n

Lk=l

~n =

as in Lemma 6.7, define

vk

Lj =1

=

~nj

An

and

Thenv

~n + ~ a.s., and moreover,

(23) n E: N ,

since the A are bounded and form a non-decreasing sequence. Supposen

that ~O E S(p~-l) Proceeding as in (20) , it is seen that there exist

S(p~-l) vsome ~n E: n E: N , with ~n + ~ , and since ~l' ~2' ... E: GS (A)n

by (23), it follows that ~O E: GS(A) .

Conversely suppose that ~O E: GS (A) , say ~ = ~l + ... + ~k forasome k E: Z and ~l' ... '~k E: SeA) Here we may assume without loss

+

that ~. ~ a for each j E: U, . .. ,k} , say that ~.f. ;t a where f. E: FJ J J J c

As M1 in Lemma 6.7 we can choose

k 1Ml = U {~ E M: ~f. >2~·f.}j =1 J J J

-85-

since by Theorem 6.5

k1

k-1 1

AMI ~ L A{~f . > '2 ~.f.} c: I AlT f. (2~·f., 00) < 00j=l J J J j=l J

J J

The set Ml being open and containing ~l' ,]1k ' we get

and hence by (23)

(24 ) n € N .

Let £ > 0 be arbitrary, and choose 0 € (0, £/2) such that

(25) p(]1, ]10) v p(]1I, 0) < 0 =? p(]1 + ]1', ]1) < £/2 ,

which is possible since the mapping (]1, ]1') + p(j1 + ]1',]1) is continuous.

Since ~ - ~ X0 a.s., we may further choose n € N such thatn

(26) P{p(~ - ~ , 0) < c} > 0 .n

By (24) - (26) and the independence of t" and':on ~ - ~n ' we obtain

= P{p(~ - ~n' 0) < o}P{p(~n' ]10) < &} > 0 ,

and since £ was arbitrary, it follows that

This completes the proof.

-1]10 € g(P~ ) , as desired.

-86-

NOTES. The canonical representations occurring in Lemmas 6.1 and 6.3

are classical, (see e.g. Feller (1968) and (1971)), while the one in Theorem

6.5 is due to Matthes (1963) and Lee (1967), (see also Mecke (1972)). The

corresponding convergence criteria are due to Kallenberg (1973a), (though

the proof of Lemma 6.1 follows that of Feller (1971) for general random

variables). As for Lemma 6.4, the first part is given in Kerstan, Matthes,

and Mecke (1974), page 49, while the second part is new. Lee (1967) gave a

proof of Lemma 6.7 along with a heuristic argument for Lemma 6.6; the

presentproof was supplied for point processes by Jagers (1973). Finally,

Theorem 6.8 extends a result for point processes by Kerstan, Matthes and

Mecke (1974), page 95, (see also Kallenberg (1973a)), while Theorem 6.9

is new.

The reader should consult Kerstan, Matthes and Mecke (1974) for

a different approach to the theory of infinitely divisible point processes,

where (16) plays the role of a basis for the whole theory while (14) and (16) fit

into their general theory of "showerfields".

PROBLEMS.

6.1. Prove the Z+-version of Lemma 6.1 directly by means of generating

functions.

6.2. State and prove tightness criteria corresponding to the convergence

assertions in Lemma 6.3 and Theorem 6.5.

6.3. dA ) n E N in Lemmas 6.1 and 6.3 and in TheoremLet F;, = l(a ,n n' n

6.5. Give criteria for convergence in distribution of {F;,n} in

terms of {a } and {A }n n

6.4. Assume that S is countable. Show that there exists some fixed

f E F such that, for any point process F;, on S with F;,S < 00

a.s., F;, is infinitely divisible iff F;,f is infinitely divisible

-87 -

vanishing constant component. (Hint: use the result in Problem

3.8.)

6.5. Let E, be a point process on S. Show that E, is infinitely

divisible iff E,f is infinitely divisible with P{E,f = O} > 0

for every f E Fc

6.6. For E, = I(O,A) on S with AM < 00 , state and prove a relationship

between the set of atom positions of pE,-l and that of A. Show

also that pE,-l can have no atoms if AM = 00 (Hint: Let f E F

be strictly positive and such that E,f < 00 a.s. Then-1ATI f R~ = A{~f > O} = AM , so the last assertion follows from the

corresponding fact for random variables due to Blum and Rosenblatt,

see Lukacs (1970), page 124.)

7. FURTHER RESULTS FOR NULL-ARRAYS.

In this section we shall see how the results of Section 6 can be

improved in certain special cases. Let us first consider the case of

random measures with independent increments, (sometimes called completely

random measures). By this we mean random measures E, such that

[,B l , ... ,E,Bk are independent for arbitrary disjoint Bl , ... ,Bk E B ,

kEN . Let us say that a measure ~ E M is degenerate if ~ = ho s

for some b;. 0 and s c S .

LEMMA 7.1. Let E, ~ l(a,A) on S. Then E, has independent increments

iff A is confined to the class of degenerate measures.

-88-

PROOF. Suppose that A has the asserted property, and consider arbitrary

kEN and disjoint Bl ,

for any t l , ... ,tk E R+

,Bk E B. Writing M' = M\{O} , we obtain

- log E exp(- L t.~B.)j J J

t.nB.J J

(1 - exp(- L t.llB.))A(dll) =j J J

= L t.aB. + L f ( 1 - exp (- t j lJ Bj )) A(dII ) =j J J j

lJB. >0J

= I {tjaB j + J (1 - exp(- t. lJB.))A(dJ1 )} =j J J

M'

= - log IT E exp(- t.~B.) ,j J J

proving that ;Bl , ... ,;Bk are independent. Conversely, suppose that

; has independent increments, and let A and B E B be disjoint. Since

a is additive, we obtain as above

(1) o .

But the function f(x) :: 1 - -xe is strictly concave on R , so+

f(x + y) < f(x) + fey) for every positive x and y, and hence (1)

is only possible provided A{J1A > 0 , )lB > O} = O. Making a partition

of S into countably many sets of diameter < € , it follows that, for

)l E M' a.e. A, at most one of these sets can have )l-measure > 0

Since € can be chosen arbitrarily small, this proves that lJ is degenerate

a.c. A.

-89-

THEOREM 7.2. A random measure ~ on S is infiniteZy divisibZe with

independent increments iff

(2) -H flog Ee = af +

R txs+

f E: F ,

for some a E: M and some y E: MeR t x S)+

with y(o x B)K < ~ , B E: B ,

and in this case a and yare unique. Moreover, any a and y with

the stated properties may occur in (2).

random measures on S and if I c B~

iff

If {~ .} is a nuZZ-arPay ofnJ

dis a DC-semiring, then I·~ . + ~] nJ

(i) \' -1 wK L . P (~ . I) + a 10

0+ Ky (. X I) ,

] nJI E: I , and

(ii) I·p{~ .1 1 A ~ .1 2 > £} + 0 for every £ > 0 and disjoint] nJ nJ

11' 12 E: 1 .

In the point process case we have a = 0 whiZe y 1.-S confined to N x S

Tn this case it suffices to take £ = 0 in (ii) and to repZace (i) by

(i) , I -1 w y (. I) I.P(~ .1) + x on N , I E:] nJ

PROOF. Suppose that ~ is infinitely divisible with independent increments.

Let M E M be the class of degenerate measures, and for ~ EM, let t be~

the unique atom position of ~. From Lemma 2.1 it is seen that M E: M and

that is a measurable mapping of M

Then (2) is

~ -> (~S, t )~

be the measure on R' x S+

induced by

into R t X S .+

A via this mapping.

Let y

obtained by transformation of the integral in (6.6), and since

(3)

-90 ..

y(dx x B) = A{~B € dx} x > 0 B € B ,

we have -1y(o x B)K = (A~B )K < m B € B . To see that y is unique,

note that the inverse mapping (xJt) + xO t transforms y into some A

and (2) into (6.6). From the above mappings it is further seen that any

a and y with'the stated properties are possible in (2).

As for the convergence assertion, the necessity of (i) and (ii)

follows easily from (3) and Theorem 6.5. Conversely, suppose that (i)

and (ii) hold. Then (6.9) follows from (i), and it reamins to prove (6.8).

It is then enough to take the components II' ... ,Ikof V in (6.8)

disjoint and to verify that

k k(4) L p n H: .r. ~ t.} + A n {~I. ~ t.}

j i=ln] 1 1 i=l 1 1

for any t = (t l , with t. = 0 or A{~I. = t.} = 0111

i = 1, ... ,k . Since events corresponding to t. = 01

can be omitted

from (4), we can assume that all the t.1

are > 0 But then (4) follows

from (i) for k = 1 and from (ii) and Lemma 7. 1 for k·~ 2. The last ,

assertion follows from the corresponding fact in Theorem 6.5.

The most general random measure with independent increments is obtained

from the one in Theorem 7.2 by addition of at most countably many fixed atoms:

COROLLARY 7.3. A random measure n on S has independent increments

iff it can be written in the foY'/11

(5)k

n = t;. + L S.Otj=l ] j

wheT'e ~ satisfies (2) faT' some

-91-

a and y with a{s} = y(R' x {s}) = 0 ,+

S E S , some k E Z and distinat t. :s; S .. j :s; k .. and some R -valued+ J +

T'andom va:fliables S. with P{S. > a} > 0 , j :s; k J whiah aT'e mutuallyJ J

independent and independent of ~ In this aase .. the above deaomposition

is unique apaT't fT'om the oT'deT' of terms.

PROOF. The sufficiency and uniqueness assertions are obvious. Next suppose

that 11 has independent increments. Like any random measure" 11 can have

at most countably many fixed atoms, i.e. I1{S} = a a.s. for all but countably

many s E S , (cf. the argument in Billingsley (1968), page 124). Let t.J

and S· ,J

j :~ k , be the corresponding atom positions and sizes, and let

~ be defined by (5). Since 11 has independent increments, it is easily

seen that the Sj are mutually independent and independent of ~,and

further that ~ has independent increments. Moreover, ~ has no fixed

atoms, so if {B .} c B is a null-array of partitions of some fixed setnJ

B E B , it follows by a simple compactness argument that {~B .}nJ

is a

null-array of random variables. Hence, by Lemma 6.1, the common row-sum

~B must be infinitely divisible. A similar argument shows more generally

that (E:Bl , ,~Bk) is infinitely divisible for any kEN and disjoint

Bl , ... ,Bk E B , and so we may conclude from Lemma 6.4 that ~ is itself

infinitely divisible. In view of Theorem 7.2, this completes the proof.

By comparison of (2) and (1.2), it is seen that a point process

~ on S is Poissonian iff it is infinitely divisible with independent

increments and with the measure y in (2) confined to the set {I} x S

-92-

In this case, ~ has intensity

E~B = yeO} x B) B E B •

Specializing Theorem 7.2 to this case, we obtain the important

COROLLARY 7.4. Let ~ be a Poisson process on S with intensity A EM,

and let {~nj } be a nuU-arTay of point prooesses on S Further suppose

that I c BA is a DC-semiring . Then I.~ . ~ ~ iffJ nJ

(i)

(ii)

2'P{~ .1 > O} + AIJ nJ

I.p{~ .B > l} + 0J nJ

IEI,and

B E B •

We are now going to consider the convergence of null-arrays towards

two special types of infinitely divisible random measures which play the

same role here as do the simple point processes and diffuse random measures

in Sections 3 - 5. But first we define regularity for canonical measures

A and for null-arrays {~nj} as follows. Given any covering class

C c B and any a > 0 , we shall say that A is a-regular w.r.t. C

if there exists for every C E C some array {Cmk } c C of finite coverings

of C such that

(6) lim 2 A{lJCmk ?: a} = 0 .]}t+OO k

The notion of a-regularity of {~nj } is defined analogeously but with

(6 ) replaced by

-93-

lim limsup L L P{~ .C k ~ a} = 0~ . k nJ mUl'- n-+oo J

"

As before, regularity means a-regularity for every a > 0 .

be a null-array of point processes on

is

S and

u c B~Further suppose that

Then \ ~ ~ E, if.'~Lj nj Ja DC-semiring.I c B~

Let {~ .}nJ

*with A{~ ~ ~ } = 0 .

THEOREM 7.5.

let [, ~ 1 (A)

a DC-ring and

(i) lim L P{E, .U > O} = A{~U > O}. nJn-+<oo J

U E U ,

(ii) limsup L P{E, .1 > I} ~ A{~1 > I} 1 < I ,n-+oo j nJ

(iii)

Moreover,

lim limsup L P{~ .B > k} = 0 B E B .k-+oo n-+oo j nJ

dLj~nj + ~ follows from (i) and (ii) if A &S 2-regular w.r.t.

I and from (i) alone if {E,nj} is 2-regular w.r.t. I or if

(7) limsup L EE, .1 ~ A~I < 00

n-+oo j nJI E I .

The proof of this theorem is similar to that of Theorem 4.8, except

that we now rely on the following two lemmas, playing the roles of Theorem

3.3 and Lemma 4.7 respectively.

LEMMA 7.6. Let A be the canonical measure of an infinitely divisible

point process, and suppose that U c B is a DC-ring and I c B a DC-

* *cerm:-ring. 7'hen A{~ ~ ~ } = 0 iff A{~1 > l} :0; A{ll 1 > l}, 1 E I ,

and in that case A &S uniquely determined by A{~U > O} U E U •

-94-

PROOF. For fixed e E U , let Ae be the measure on N' = N\{O} induced

by A via the mapping ~ + e~ , i.e.

MEN n N' .

Then

so either this quantity is zero, or Ae can be normalized to a probability

measure on N' , i.e. to the distribution of some simple point process.

Hence, by Theorem 3.3, I.e is completely determined by the quantities

U € U ,

Le. by all U E U But AC

determines the measure

for any f E F with support in C so by Theorem 6.5, the family {AC}

determines A. We may now complete the proof as in case of Theorem 3.3.

LEMMA 7.7. Let {E,;nj} be a nuZZ-array of random measures on S , and

thatd Further suppose that d

I (a,A) andsuppose I.E,; . + som~ n· E,; =J nJ

that U c B '[.,s a DC-ring satisfying

(8)-tE,; .U -tnuliminf I (1 - Ee nJ ) ~ taU + 1.(1 - e )

n-+oo jU EO U ,

for some t > 0 Then nC = 0 a.s. for every compact C E B with

~C = 0 a.s. In the point process case, (8) can be repZaced by

1iminf I P{~ .U > O} ~ A{~U > O}n-+oo j nJ

U t: U .

THEOREM 7.8.

on S and let

-95-

This may be proved in the same way as Lemma 4.7.

We now turn to the analogue of Theorem 4.9.

Let nnj} be a nuU-arTau of point processes (random measures)

~ ~ I (a,A) with a = 0 and A{}l;t /} = 0 (or' with AMc = 0d

respectively). Further suppose that sand t are fixed real numbers with

semi-ring) .

a covering class (DC-o < s < t

(i)

(ii)

and that U c BI; is a DC-ring and I c BI;

Then \ I; ~ I; if.fL.j nj

-tl; .U -twlim L (1 - Ee nJ ) = taU + A(1 _ e U)n-+oo j

-sl; .1 -STI

limsup r (1 - Ee n J ) ~ sal + A(l _ e I)n-+oo j

U E U ,

I € I ,

I E I .(iii)

Moreover.,

lim limsup I P{~ .1 > t} = 0t+oo n+oo j nJ

r·t; . ~!; follows from (i) alone if {!;nJ'} ~s 2-regular (regular)J nJ

w.r.t.

(9)

I or if

limsup I EI; .1 $ aC + ATII

< 00

n-+oo j nJI E I .

PROOF. We confine our attention to the random measure case, since the

point process case is similar but simpler. Suppose thatd

\.1; . + !;L. J nJ

and let B E U or I Then L.I; .B &!;B by Lemma 4.3, so by Lemma 6.1J nJ

This implies (i) and (ii), while (iii) follows by Lemmas 4.6 and 6.2.

-96-

Conversely, suppose that (i) - (iii) hold. Since -tx(1 - e )/x

is a decreasing function of x > 0 we have

and so we get for any r > 0 and U E U

o :-:; x :0; r ,

By Lemmas 4.6 and 6.2, it therefore follows from (i) and (iii) that {LjSnj}

is tight. But in that case every sequence N' c N has a subsequence Nil

by Lemma 7.7, it follows from (i), (ii) and

dsuch that Los, + some n

J nJ

6.5, and since U u I c Bn

(n E Nil) Now dn = some by Theorem

the necessity part of the present theorem that

-t7T -t7TtaU + ~ (1 e U) = telU + 1.(1 e U) U E U ,

-S7T -S7Tsal + ~(l - e l) :-:; sell + 1.(1 - e l) I E: I ,

which by Theorem 6.5 is equivalent to

(10) -tnU -tsU U E U ,Ee = Ee

(11) -snl -ssl I E: IEe ~ Ee

Furthermore, it follows from Theorem 6.8 that s is a.s. diffuse, possihly

apart from atoms of fixed size and location which arize from the atoms of

(1. Let us now consider any fixed XES Choosing Bl , B2, ... in U

or 1 respectively such that Bn ~ {x} , it follows by monotone convergence

-97-

from (10) and (11) that

(12) Ee-tnJx} Ee -tHx} -ta{x}= = e

(13) Ee-sn{x} :<: Ee-s~{x} -sa{x}= e

If n{x} were non-degenerate, it would follow from Jensen's inequality

for strictly convex functions that

(e -sa{x}) tIs = -ta{x}e

which is impossible. Hence n{x} must be degenerate, and by (12) we

obtain n{x} = a{x} a.s. Writing a' for the purely atomic part of

a , it follows that n:<: a' a.s., and since (10) and (11) may be written

in the form

(14 )-t (n-a')UEe = Ee-t(~-al)U U E U ,

Ee-s(n-a')I :<: Ee-s(~-al)I I E I ,

it from Theorem 3.4 that ~ ... a' d - a' i.e. that dis seen n , t;. = n

This shows that Ij~nj ~ ~ (n E Nil) , and since N' was arbitrary, even

that Lj t;.nj ~ t;. (n E N) , proving the first assertion.

The assertion involving regularity of {~nj} follows by the usual

arguments, so it remains to prove the assertion involving (9). In this

case we get by Fatou's lemma

(15) EnI :S liminf L E~ . ImiN" j nJ

:S limsup L E~ .1 ~ aI + AnI = E~In-+oo j nJ

I E I ,

-98-

replacing (11), so (13) is now replaced by

(16)

If n{x} were non-degenerate, we would obtain from (12), (16) and Jensen's

inequality

-ta{x}~ e ,

which is impossible, so again we get n{x} = a{x} a.s., x € S , proving

that n ~ a' a.s. Thus it follows by (14), (15) and .Theorem 1.4 that

~ - a' g n ~ a' , and the proof may be completed as before.

NOTES. The canonical representation in Theorem 7.2 and Corollary 7.3,

which is a well-known classical result for random measures on R, is

due in the present generality to Kingman (1967). The present approach

via Lemma 7.1 a~ well as the convergence criterion in Theorem 7.2 were

given in Kallenberg (l973a). Corollary 7.4 is a classical reSUlt, proved

'" "in particular cases by Palm (1943), Hincin (1960) and Ososkov (1956), and

in general (on R) by Grige1ionis (1963), (see also Goldman (1967) and

Jagers (1972)). Finally, Theorem 7.5 is taken from Ka1lenberg (1973a)

while Theorem 7.8 is new.

PROBLEMS.

7.1. Show in analogy with Lemma 6.6 that the random measure ~ in Theorem

7.2 is distributed like a + Jxotn(dxdt) , where n is a Poisson

process on R' x S with intensity y+

7.2.

7.3.

7.4.

-99-

Let {~nj} be a null-array of point processes on S. Show that

I.I;. is tight with only Poissonian limit distributions iffJ nJ

\.P{I; .B > O} is bounded and \.P{c .B > I} + 0 for every B E B .L J nJ L J snJ

State and prove a similar tightness criterion corresponding to the

convergence assertion in Theorem 7.2.

Prove Corollary 7.4 directly from Theorem 4.8 and Lemma 6.1 in the

particular case when A E Md

7.5. Prove Theorem 7.8 in the point process case. Show that I may

be assumed to be a covering class even in the random measure case,

provided a is known to be diffuse.

7.6. Let ~ ~ lCa,A) on S , let C c B be a covering class and let

a > 0 Show that A{ll ;t O} = 0 if A is a-regular w.r.t. Caand that the converse is also true provided C is a DC-semiring

and E~ E M

7.7. Let {~nj } be a null-array of random measures on S such that

I.~ . §. some *I (a, A) Show that A{ll ;t O} = 0 if {~nj } isJ nJ E:

a-regular w.r.t. B~ for every a > E: , and that the converse is

also true provided E~ E M.

8. COMPOUND AND THINNED POINT PROCESSES.

The notions of compounding and thinning were introduced in Section

1, and in Corollaries 3.2 and 3.5 we proved two uniqueness results for

compound point processes. The aim of the present section is to study

the convergence in distribution of such random measures, and we shall

-100-

first consider a case when the limits are themselves compound point processes.

THEOREM 8. 1• For n E Z+ ' let i:n be a 8 -compound of nn . Thennd

andd imply that

dConversely" suppose thatSn -+ 130 nn -+ nO ~n -+ ~o

d and that {nn} is tight. Then ~ is a So-compound ofE, -+ Bome E,n

no for some 80 and nO ' and if {Cnn } is 2-reguZar w.r.t. B~ for

some C E B with l;C ~ a"

there exist some constants PO' PI' ... E (0,1]

with inf 0 such thatd

8 ' and d , where ~p = Pn > 13' -+ n' -+ nO " 1-S a Pn-n 0 n

thinning of nn' n E Z+ ' while

(1) n E Z+

PROOF. Let us first suppose that

be a 13n-compound of nn' n E Z+

is then equivalent to

sn ~ 80 and ~ ~ nO ' and let l;n

dBy (1.5) and Theorem 4.5, l;n -+ £'0

(2) E exp(nnlog ~n 0 f) -+ E exp(nOlog ~O 0 f) f E Fc

where n E Z+ ' and by continuity and boundedness, (2) will

follow if we can prove that

(3) f E Fc

By Theorem 5.5 in Billingsley (1968), it 1S enough to prove (3) for non-

random nn' i.e. to show thatv

lln -+ 110 implies

(4) f E Fc

Considering a fixed

-101-

f E F ,we may assume thatcw

]In -+ ]JO ;t 0 (cf. Problem

4.5), and after a suitable normalization that the ]In are probability

measures. But then (4) may be written in the form

for some random elements in S with Since

f is bounded, the sequence {log </> 0 f}n is uniformly bounded, and hence

(5) is a consequence of the relation

(6)

Applying Theorem 5.S in Billingsley (1968) once more, it is seen that

the quantities Tn = sn may be considered as non-random with sn -+ So '

and since in that case xn = f(sn) -+ f(s) = x , it remains to prove that

whenever But this follows from the fact

that the convergence </>n -+ </>0 is uniform on bounded intervals.

Let us now suppose conversely that s ~ s and thatn

is tight.

Since 0 E M is a O-compound of 0 E N , we may assume that s ~ 0 Let

the sequence N' c N be such thatd (n EN') If

d0nn -+ some n n ,

we obtain for any fixed B E B

so by (1.5) and bounded convergence

d~ n B -+ 0 ,

n

-s BEe n = E exp(n B log </> (1)) -+ 1 ,

n n

-102:-

and hence F,B ~ 0, B E B , yielding the contradiction ~ = 0 a.s.

Thus n ~ 0 .

We shall use this fact to show that {Sn} is tight or, what amounts

to the same, that

(7) lim limsup (1 - ~n(t)} = 0 .t-+O n-+oo

Suppose on the contrary that the limit in (7) is greater than some E > a .

Then there exist arbitrarily small numbers t > a such that, for some

sequence N" eN' ,

(8) 1 - ~ (t) > En n E N" .

Choosing B E B~ n Bn

such that nB ~ 0 and noting that

(9) 1 - x ~ - log x ~ 2(1 - x)

it follows by (1. 5) and (8) that

-tl; B -En Bn E exp(nnB log ~n(t)) E exp[-nnB(l - ~n(t)}] Ee nEe = ~ ~

and I; B ~ I;B andd

obtain -t!;B -EnBsince ~B-+ n.B , we Ee ~ Ee Lettingn

A h th d·· 1 Be-En.B < 1 . h (7)·t -+ , we reac e contra lctl0n ~ , provIng t at IS

indeed true. Thus the sequences {Sn} and {nn} are both tight, and

in particular we may consider convergent subsequences and use the direct

part of the theorem to conclude that I; has the asserted form.

Let us now suppose that {C~} is 2-regular w.r.t. BI; for some

C E B~ with I;C ~ 0 • and assume that S ~ S' and n i n' as n -+ 00

'-> n n

-103-

through N' c N Since E; is then as' -compound of n ' , and moreover

S' ~ 0 it follows from (1. 5) that B = BE; and so Cn' must ben 'simple (cf. Problem 4.9) . Put P = P{S' > O} , let nO be a p-tFtinning

of n' and let So be an R+~valued random variable satisfying

(10) -1P{SO > x} = p P{S' > x}

Then ¢ = LS

and cP ' = LS' satisfy the relation cPt :: P<PO + (1 - p) ,00

so letting E;O be a So -compound of nO and using (1. 5) and (1.6), we

obtain for any f E F

LF,; (f)o

= L (-log cPO 0 f) = Ln , (-log[lnO

pel - <Po 0 f)J) =

= L ,T-log[l - (1 - <P' 0 f)]) = L ,(-log <p' 0 f) = Lt"(f) ,n n ~

and hence F,; g F,; by Theorem 3.1.o Now P{So > O} = 1 , and moreover

is simple by Problem 2.4, so it follows from Corollary 3.5 that

and are uni~uely determined by pF,;-l

Choose any fixed y > 0 satisfying N80 > y} > 0 and NsO = y} = 0 ,

and define p = P{s > y} , n E Z+ Then PO > 0 , and moreovern n

liminf p > () since if Pn -+ 0 as n -+ ClO through some sequence N' c N ,n n

we could choose some further subsequence Nil c N' such thatd

8n

-+ some S' ,

and then (10) would hold with p = P{8' > O} > 0 , and we would get the

contradiction

P = P{S > y} -+ P{S' > y} = P{B' > O}P{SO > y} > 0n n

(n E Nil) .

-104-

To obtain p = inf p > 0 , we may replace the finitely many zeros inn

the sequence {Pn} by arbitrary positive numbers. We can now let ~

be a Pn-thinning of nn' n E Z+ ' and define random variables 8 'n

satisfying (1).

Given any sequence N' eN, we may choose a subsequence Nil c N'

In this case the constant

such that d8 + somen 13' ,

p

dn + some n' andn

in (10) equals

p + p' = P{ 13' > y} (n E Nil) .n-1

p'PO ,and in particular

Using the direct part of the theorem, it is seen-1= p'PO

dNil n' +, 'Inthat, on no ' a p'-thinning of n' . On the other hand, it

was shown above that n is a (p' po1)-thinning of n' , and so nO is

dlike n' a po-thinning of n , and we get n' + n' (n E Nil) . Furthermore,o n 0

it is seen from (1) that 13' ~ 8" (n E Nil) , wheren 0

P{13' > O}

-1 -1 -1P{I3H > x} = pp' P{I3' > x} = pp' p'Po P{8

0> x} = -1

PPO P{13 0 > x} =

= N130 > x}

so 13" ~ 13' and it follows thato 0 13' ~ 8'n 0

(n E Nil) . In view of Theorem

2.3 in Billingsley (1968), this completes the proof.

We are now going

compounding variables

to consider the entirely different case when the

dI3n satisfy 8n + O. The class of possible limits

may then be described as follows.

Consider arbitrary a E R and A E M(R' ) with AK < 00 , and define+ +

-IT

(11) 1/J (t) = at + A(1 _ e t) t E R+

-105-

By Theorem 7.2, there exists for every Y E M some random_measure ~

with homogeneous (w.r.t. y) independent increments satisfring, for

any f E F ,

log Ee -~f = oJ.lf + J (l - e -Jef (t)) (A x )l) (dJedt) =R'xS

+

= t {af(t) + JR

, (1 - e-xf(t))A(dx)}y(dt) = )l(t/J 0 f) .

+

By Lemma 1.4, we may consider )l = n as a random measure and mix w.r.t.

its distribution, thus obtaining

(12) L (t/J 0 f)n

f E F .

Note in particular that, if a = 0 and A = 01 ' then ~ is a Cox process

directed by n. We shall need the following uniqueness result which is

similar to Corollary 3.5.

LEMMA 8.2. Let ~ be a random measure on S satisfying (11) and (12)

for some n ~ a and A ~ and suppose that a + AK = 1 whiZe Cn is

diffuse for some C E B wi th nC ~ 0 •

uniqueZy determined by p~-l.

PROOF. By (12),

Then -1Pn ~ a and A are

B E B ,

and replacing B by B n C , it follows by Theorem 3.4 that P(Cn)-l

is uniquely determined by p~-l. But then it follows as in the proof

-106-

of Corollary 3.5 that even ~ is uniquely determined, and by Lemma 6.1,

~ determines a. and A . The uniqueness of -1Pn now follows as in

case of Corollary 3.5.

To avoid trivial complications, let us assume that any compounding

dvariable B is such that f3 ~ 0 , and further that the a. and A in

(11) are normalized in such a way that a. + AK = 1 .

THEOREM 8.3. For n E N ~ let ~n

Suppose that Bn

i 0

be a Bn-compound

Then ~ 5! 0n

of ~ and put

"'f'+' d 0v J c n +n n

Purthermore~ the conditions

(i)

(ii)

dc n + n ,n n

imply thatd

~ + somen ~ satisfying (11) and (12). ConverseZy~

implies that ~ satisfies (11) and (12) for some n ~ a. and A ~ and

if moreover

then -1Pn ~

{c Cn }n n

a. and

is regular w.r.t. B~ for some C E B~

A are unique and satisfy (i) and (ii).

with ~C ~ 0 ,

PROOF. Writing ~ = Ln Bn

n EN, we get as before

-t~ B(13) Ee n = E exp(nnB log ~n(t)) B E B n EN.

Now B ~ 0n

large n E N

implies that ~ + 1 , so by (9) we get for sufficientlyn

-107-

and it follows from (13) that

2cn

-~ BE exp(-2cnnnB) ~ Ee n ~ E exp(-cnnnB) B E B ,

showing that

assertion.

r B ~ 0~n

iff dc n B + 0 •n n B E B • This proves the first

Next suppose that (i) and (ii) are satisfied. Proceeding as in the

proof of Theorem 8.1, we obtain ~ ~ ~ with p~-l determined by (11)n

and (12), provided we can show that

whenever x + x .n

But this follows from the

fact, essentially being proved in Lemma 6.1, that -1-cn log ~n tends to

W uniformly on bounded intervals.

Suppose conversely that ~n ~ some ~, and let B E B~ be arbitrary.

Then !; B ~ ~B by Lemma 4.3, so by (13)n

and since this limit tends to one as t + 0 , there exists for each E > 0

some t E (0,1) satisfying

(14) E exp(n B log ~ (t)) ~ 1 - E/2n nn EN.

Now it follows from (9) and the elementary inequality

that

1 -

-108-

-tx xe ~ tel - e- ) t E [0,1]

-tS- log ~ (t) ~ 1 - ~ (t) = E(l _ e n} ~n n

'" '"and so by (14) and Cebysev's inequality

-6tE (1 _ e n) = tc

n

P{ c n B >-1 2} P{tc n B > log 2} P{-n B log ~ (t) > log 2}t log = ~n n n n n n

= PO - exp(n B log ~ (t)) > !.} ~n n 2

~ 2EO - exp(n B log ~ (t))} ~ £ ,n n

proving tightness of {c 11 B}n n Since B E B~ was arbitrary, it follows

by Lemma 4.6 that {c n }n n

is tight.

Let us no\'l assume in addition that ~ 2 o. Proceeding as in the

proof of Theorem 8.1, we can then prove that

(15) lim limsup c-1(1 - ~ (t)) = a .t~O n+oo n n

V \IBy Cebysev's inequality, we further obtain for any r,t > a

-S tc-lE(l-e n) c-l(l_~ (t)}

> r} ~ _n = _n n _l_e-rt l_e-rt

and letting in order n +00 r ~ 00 and t ~ a , we get by (15)

lim limsup c-lp{S > r} = a ,n n

r~oo n~oo

which shows that the sequence {c-lK(pS- l )} of probability measures onn n

R is tight. In particular there exist some sequence N' ~ N and some+

-109-

n, a and A such that (i) and (ii) hold on Nl , and we may conclude

from the sufficiency part of the theorem that ~ has the asserted form.

(In the case ~ ~ 0 we may take n = 0 and choose arbitrary a and

A .)

Let us finally assume that is regular

w.r.t. B~ for some C E B~ with ~C 2 O. Since the sequences {cnnn}

and {c-lK(PS- l )} are both tight, any sequence N' c N must contain an n

subsequence N" c N' such that (i) and (ii) hold on N" for some n,

a and A But then ~ satisfies (11) and (12), and in particular

B~ = Bn

' so it follows by Problem 4.9 that Cn is a.s. diffuse. We

may thus conclude from Lemma 8.2 that -1Pn » a and A are uniquely

determined by -1p~ ,and hence, by Theorem 2.3 in Billingsley (1968),

that (i) and (ii) remain true for the original sequence. This completes

the proof.

The above convergence criterion takes a particularly simple form

in the case of thinnings:

COROLLARY 8.4. For n EN, let Pn E (0,1] and let ~ be a p -thinningn n

of some point process Suppose that P ~ 0 •n

Thend

~ ~ somen

iff P n i some n, and in this case ~ is a Cox process directed by n.n n

PROOF. The preceding proof applies with cn - Pn ' a = 0 and A = 61

(and it may even be simplified in the present case, since eii) is auto-

matically satisfied). No regularity assumption is required on account

of Corollary 3.2.

-110-

The last result leads to an interesting characterization of the class

of Cox processes:

COROLLARY 8.5. Let ~ be a point process on S. Then ~ is a Cox

process d-&l'ected by some random measure n iff ~ is for every p E (0, IJ

a p-thinning of some point process ~. In this case np

is a Cox process

directed by -1P n.

PROOF. Let p E (0,1] , let ~ and np be Cox processes directed by

and -1 respectively, and let ~p be a p-thinning of By11 P 11 np

(1. 4) and (1. 6) we then obtain for any f E F

= L (-log[l11p

p-thinning of some

so

L~ (f)P

pel - e-f)]) = L -1 (1 - [1 - pel - e-f )]) =

p n

= L -1 (p(l - e-f)) = L (1 - e-

f) = L~(f) ,

P n n ~

Conversely, suppose that ~ is for every p E (0,1]

-1Applying Corollary 8.4 with p =nandn

a

~n = ~ , n EN, it is seen that ~ must be a Cox process.

We conclude this section with a useful criterion for infinite divisibility.

LEMMA 8.6. Let ~ be a random measure on S and, for any c > 0 , let

11C

be a Cox process directed by c~. Then ~ is infinitely divisible

iff 11 c is infinitely divisible for every c > 0 .

PROOF. The necessity part is obvious. Suppose conversely that nc is

infinitely divisible for every c > O. Put p~-l = D and introduce

-111-

in the spaces of distributions on Nand M t~e operators n Vp P

and J corresponding to division by p , p-thinning and formation of

Cox processes respectively, p E: (O,lJ . From (1.4) and (1.6) it is seen

that V and J commute with convolution. Using this fact and the lastp

assertion in Corollary 8.S, we obtain

(V (In D)l/n)n = V In D = JDP P P P

P E: (0,1] n EN.

Thus (JD)l/n is a p-thinning for every p E: (0,1] and nE: N , so by

Corollary 8.5 there exists for every n E: N some distribution Dn on

M such that UD) lIn = JDn

But then

n E: N ,

and it follows by Corollary 3.2 that

o is infinitely divisible.

D = (D )nn n E: N , proving that

NOTES. Theorem 8.1 is new while Theorem 8.3 and Corollary 8.4 were given

in Kallenberg (1975). The latter result extends some classical work of

Renyi (1956), Nawrotzki (1962), Be1yaev (1963) and GOldman (1967), who

all consider p-thinnings of some fixed point process and change the "time"

scale by the factor p-1 to be able to obtain convergence. Finally,

Corollary 8.5 is due with a direct proof to Mecke (1968 ) and (1972). (ef.

Kerstan, Matthes and Mecke (1974), page 318), while Lemma 8.6 is due to

Kummer and Matthes (1970).

-112-

PROBLEMS.

8.1. State and prove the thinning case version of Theorem 8.1. Note

that no regularity condition is required in this case.

8.2. Let f, be a 8..f,Qmpound of s.ome simple point process*

p '" P{B > O} Show that E;O+ is then a p-thinning

this fact to give a new proof of Corollary 3.5.

n and put

of n. Use

8.3. For n E N , let E; be a B -compound of nn , and suppose that

B ~ 2 0n n d d

some 8 Show that E; + some E;, iff nn + some n ,n nand that E;, is then a B-compound of n State and prove the cor-

responding version of Theorem 8.3.

8.4. Let E;, be a B-compound of n and suppose that 8 is infinitely

divisible. Show that E;, has then infinitely many essentially dif­

ferent representations as a 8'-compound of n' with 8' > 0 a.s.

(Hint: Replace n by nn, n E N Proceed as in the proof of

Theorem 8.1 to reduce to the case 8 > 0 a.s. Finally prove

that the representations thus obtained are really different.)

8.5. Let E;, be a 8-compound of n and suppose that n is infinitely

divisible. Show that E; is then also infinitely divisible. Prove

the corresponding result for the random measures defined by (11)

and (12).

8.6. Let E;, be a Cox process directed by n or a p-thinning of nShow that it is possible that E;, is infinitely divisible while

n is not. (Hint: It is enough to consider Z+-va1ued random variables.

To construct our n , we may start with the familiar Raikov-L~vy

example reproduced in Lukacs (1970), page 251, and add a suitable

infinitely divisible variable to make the thinning or Cox variable

E;, infinitely divisible.)

-113-

9. SYMMETRICALLY DISTRIBUTED RANDOM MEASURES.

Let w E Md be fixed. We shall say that a random measure s is

·e.-

symmetricaZZy distributed with respect to w if the distribution of

kEN , only depends

If wS E R' and if T is a random element in+

S with distribution w/wS, then this property is obvious for the point

process o , and since the property is persistent under addition of in­T

dependent random measures and under mixing, it is shared by all mixed

Poisson and sample processes which are defined w.r.t. w

any such process must satisfy

In particular,

(1) P{sB = O} HwB) B E B ,

for some non-increasing function <p. If s is a mixed Poisson process

w.r.t. w with mixing random variable a , then

(2)-awB

P{sB = O} = Ee = ¢(wB) B E B ,

so in this case ¢ = L On the other hand, if wS E R' and if sa +

is a mixed sample process w.r.t. w/wS with mixing variable 'V , then

(3) P{~B O} = F. (I -wB)'V _

1/J (1 - ~~) = <PewB) B E BwS - wS ,

where 1/J is the generating function of 'V , so in this case

(4 ) <Pet) = 1/J(1 - w~) , t E: [0, wS)

1/J(t) = <t>(wS(l - t)) t E (0, I] .

-114-

rf ~ is a mixed Poisson or sample process, then p~-l is easily seen

to bc dctermined by w and ~. We shall use the symbol M(w,~) to

denote such a process ~. (Note, however, that the pair (w,~) is not

uniquely determined by p~-l since it can be replaced by (aw, ~('/a))

for any a > 0 .)

It turns out that every symmetrically distributed simple point process

is a mixed Poisson or sample process. We shall even be able to prove the

following stronger result.

THEOREM 9.1. Let ~ be a simpZe point process on S and Zet w E Md .

Further suppose that U c B is a DC-ring and that t E (O,~] is fixed.

dThen ~ = some M(w,~) iff there ex~st8 some non-increasing function

~t satisfying

(5 )

In that case

(5' )

-t~UEe = ~t CwU) U E U .

X E [0, wS] .

PROOF. Let t E R' and B E B+

and suppose that ~ is a mixed Poisson

process with mixing variable a. Then

On the other hand, we get in the mixed sample case, on account of (4)

-115-

Thus (5) holds in both cases with qit defined by (5'). For t", 00

this statement was proved in (2) and (3).

Suppose conversely that (5) holds, and let us first assume that

.-

t = 00

w ~ 0

Since w = 0 clearly implies ~ = 0 a.s., we may assume that

Our first aim is to extend (5) to B, i.e. to prove (1). For

this purpose, let sl' 52' ... E S be a bounded sequence of distinct

points belonging to the support of w, and consider for fixed s = sn

some sequence of sets Uk E U with kEN ,satisfying Uk ~ Is} .

Then 0 < wUk ~ w{s} = 0 while ~Uk ~ ~{s} a.s., so we get

Hence by Fatou's lemma

so qi(O+) = 1. To extend this continuity at 0 to (uniform) continuity

on [0, wS) (or even on [0, wSJ if w has bounded support), let £ > 0

be fixed and choose a > 0 such that 1 - qi(30) < £. For any s,t E [0, wS)

with s < t < s + cS , we can choose sets lJ,V E U with U c V and such that

(6 ) s - cS < wU S S < t < wV < t + 0 .

In fact, let C E U be such that wC ~ t and let {C .} cUbe a null-arraynJ

of partitions of C

for construction of

-116-

Then max wC . + 0 since w E Md ' so it sufficesj nJ

U and V to consider an"n E N' with max wC . < Qj nJ

From (5), (6) and the monotonicity of ~ we get

~(s) - ~(t) ~ ~(wU) - ~(wV) = P{~U = O} - P{~V = O} = P{~U = 0 , ~V > O} ~

~ P{~ (V\U) > O} = 1 - ~ (w(V\U)) =

= 1 - ~(wV - wU) ~ 1 - ~(3Q) < E •

We may now use the monotone class theorem to extend (5) to B.

For arbitTary n E Nand h < wS/n , it is possible to choose disjoint

sets Bl , ... ,Bn E B with wBI = ••• = wBn = ~ It. fact, let C E B

with wC > nh and let {C .} be a null-array of nested partitions ofmJ

fixed k > choose-1

C For 0 we can m so large that max wCmj < kj

and then there exist disjoint unions Bkl , ... ,Bkn of the partitioning

C such that h-1

~ h j 1,sets CmI' m2' ... - k < wBkj , = ... ,n

The sets Bkj may clearly be choosen non-decreasjng in' k for each

j , and in that case the sets

desired property.

B. = U BkJ

.J k

j = 1, ... ,n , have the

With this choice of Bl , ... ,Bn ' it is seen from (1) and Lemma

5.7 that

?: 0 ,

whichSmcans that the function ~ is completely monotone (cf. Feller (1971),

together with the corresponding inequality for arbitrary

pp. 223,439).

5

If wS = ~ , it follows that is the L-transform of some

x

-117-

R -valued random variable, and we get+ . by (2) and Theorem

-e

of some Z -valued random variable, and again+

3.3. On the other hand, if wS < ro then the function ~(t) = ~(wS(l - t))

t E (0,1) , being absolutely monotone, is the probability generating function

d~ = M(w,~) , now by (3) and

Theorem 3.3. This completes the proof in the case t = ro

In the case of finite t, let ~ be a p-thinning of ~,where

p = 1 --t

e As in the proof of Theorem 3.4, we obtain

U E U ,

and it follows as above that dn = M(w, ~t) . But then n is symmetrically

distributed w.r.t. w, and from (1.6) it is easily seen by analytic con-

tinuation that this is also true for ~. In particular, (1) must hold

for some ~,and it follows as above that

now complete.

The proof is

COROLLARY 9.2. Let ~ be a point process on S and let w EM. Then

d M( ) -:f.'+' B d M( .) f B B~ = some w,~ v J ~ = some wB' ~B or every E . We may

then take wB = Bw , in which case ~B becomes the restriction of ~

to [0, wB] . Furthermore, p~-l is then uniquely determined by P(t,;B)-l

for any fixed B E B with wB > °Note that the only if part of the first assertion contains a well-

known classical property of the Poisson process.

PROOF. Let us first suppose that w E Md' Then the first assertion

is an immediate consequence of Theorem 9.2, while the second one follows

-118-

by comparison of (Z) and (3). As for the last assertion, note that ~B

has probability generating function ~(wB(l - t)}, t E [O,lJ Thus

P(~B)-l determines ~ on [0, wB] and hence, by the uniqueness of analytic

continuations, on [0, wS)

The proof for general w E M requires randomization.

variables which are uniformly distributed on

and choose independently of {T.}J

a sequence {n.}J

[0,1]

Write ~ = L is. T.J J

of independent random

It is then easily

seen that ~ = L is is a point process on S x [0,1] and that. (T.,n.)J J J

Add~ = M(w x A,~) iff ~ = M(w,~) , where A denotes Lebesgue measure on

[0,1],..

Since w x A is diffuse, the assertion is true for ~, and

so it holds for ~ as well.

We next indicate by an example how Theorem 9.1 can be used to improve

Theorems 3.3 and 3.4 in the particular case of Poisson and sample processes.

COROLLARY 9.3. Let w E Md and~ for wS = 00 or < 00 ~ let ~ be a

Poisson or sample process respectively with intensity w

that n is a simple point process on S satisfying

Further suppose

P{nU = O} = ~(wU) U E U ,

for some DC-ring U c B and some non-increasing function ~ ThBn

dE;, = n iff theY'e exist two sets Bl , Bz E B with 0 < wBl < wBZ satisfying

(7) P{~B. = O} = P{nB. = O}J J

j = 1,2 .

PROOF. By Theorem 9.1 we have

-119-.

d .n = M(w,~) In the case wS = 00 we

get by (2) and (7)

(8) Ee -as -s= e Ee-at -t= e

where s = wBl and t = wB2 while a is a random variable with L-transform

~. If a were non-degenerate, it would follow by Jensen's inequality for

strictly convex functions that

-atEe . -t= e

which is impossible and proves that a is indeed degenerate. From (8)

we get a = 1 a. s., and so ~ (x) ==-xe which means that n is a Poisson

process with intensity w. A similar proof applies to the case wS < 00 •

We are now going to characterize the class of arbitrary symmetrically

distributed random measures. Consider an infinitely divisible random

measure ~ with homogeneous w.r.t. w independent increments such that

(9) -H f- log Ee = awf +

SxR'+

(1 - e-xf(s))(w x A) (dsdx) f E F ,

where a E R and A E M(R') with AK < 00. By Fubini's theorem, the+ +

integral Is (1 - e-xf(s)}w(ds) is a measurable function of x, and

therefore the right-hand side of (9) is jointly measurable in a and

A Hence, by Lemma 1.4, we may perform mixing with respect to a and

A considered as random elements. This gives the most general distribution

in case wS = 00. The case wS < 00 is entirely different:

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THEOREM 9.4. Let W E Md\{O} and let ~ be a random measure on S .

Then ~ is symmetrically distributed w.r.t. w iff

(i) in the case wS = 00 ~ there exist some R+-valued random variable

a and some random measure A on R' with AK < 00 a.s. such+

that~ given a and A ~ the conditional L-transfoY'm of ~

is given by (9);

(ii) ~n the case wS < 00 , there exist some R+-valued random variable

is conditionally distributed

a and some point process S =

a.s. such that~ given a and

L'Oo] ..,.]

S ~ ~

on R I with+

L.S. < 00

] ]

as aw' + L.S.o ~ where WI = w/wS while T I , T 2 . are] ] T.

Jindependent random oZemp.nts in S with distribution WI

The random elements (a,A) and (a,S) respectively are a.s. unique

measurable functions of ~ .

We shall call (a,A) or (a,S) respectively the canonical random

elements of ~.

PROOF. The sufficiency of (i) and (ii) is obvious. Conversely, suppose

that ~ is symmetrically distributed w.r.t. w, and let us first assume

that wS < 00 , say that wS = 1 The symmetry assumption then extends

by monotone convergence to B and in particular it it seen that I;;S < 00

a.s. Hence by Lemma 2.1, f, has the representation

(10)

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for some random elements sd E Md , 81 ~ 82 ~ ... ~ 0 and '1' '2' '" E S , <

the latter being a.s. distinct.

are taken in random order.assume that the corresponding ,.J

Whenever two or more B.J

coincide, we may

We shall first consider the case when S is the unit interval [0,1)

while w is Lebesgue measure. Using Theorem 3.1, it is easily seen that

Ps-l is symmetric in the sense that sT- l ~ s for every w-preserving

bi-measurable 1 - 1 mapping T of [0,1) onto itself. Since a = sdS

and 81, 82 , ... are invariant under such transformations, the conditional

distributions of S ,given (a, 81, 82, ... ) , are again symmetric, so we

may consider a, 61

, 82, ... as constants. Since sd is by Lemma 2.1 a

-1 -1 dmeasurable function of S , we have for any T as above sdT = (sT )d = sd '

so even Psdlis sYmmetric. Moreover, the process sd[O,t) - at, t E [0,1] 4It

is clearly a.S. uniformly continuous and of bounded variation so, writing

Inj = [(j - l)/n, j/n) and 'nnj = Sd1nj - o./n, j = 1, ... ,n, n EN,

we get

I Tln2

J' ~ 0: ITl . I) max In. I ~ °. . nJ . TIJJ J J

a. s. , n~oo.

From the symmetry and the obvious relations

we hence obtain by bounded convergence

and

2= E I Tl nj ~ °,

j

and so

K 2

~x E{jIl nnj} = m~X{kEn~l + k~ - llEnn1 "n2} ,~ nETl~l + nen - 1) !ETlnl Tln2 1 ~ 0 .

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But this implies ~d[O,t) = at a.s. for every rational t E {O,l] ,

and therefore t;d = etA a.s.

Next suppose that 61> Q , and let v = v (~) be defined for fixed

n E N by '1 E I For k,m E Z (mod n) , let Tkm

be the transformationnv

of [0,1) which maps Ink onto In,k+m and In,k+m onto Ink ' and put

t;nm t;T-lBy symmetry we get for any A E Mm

n n

P{~T~-1P{t; E A} = L P{t; E A, \/ = k + m} = L E A, \/(t;Tkm

) = k + m} =k=l k=l

nP{t;T~

nP{I;T- l= L E A, \/ CO = k} = L E A, v(1;) = k} =

k=l k=l\/m

= P{t;T-1E A} = P{I; E A} .

\1m nm

(Note that this relation is even true when 81 = 82 ' although v is

not uniquely determined by t; in this case.) This proves that

for any nand m.

dt;nm = I;

For fixed s E (0,1] we now define t;s as the random measure obtained

from t; by replacing '1 by '1 + s (mod 1) Let m,n ~ 00 in such a way

that min ~ s , and suppose that '1 + s ~ 0 (mod 1)

for any interval I with s ¢ I , and hence also when

Then !; I ~ t; Inm s

s E 10 since

By Fubini's theorem, this implies for any

k = sup{j: 8. > O} ,J

Thereforesnm[O,l) = t;[0,1) =

d d .S =!; ~ t; , l.e.nm s

measurable function

I;s[O,l)

t;s ~ t;

kf: R+ ~ R+ '

wt;nm ~ t;s a. s. , and we get

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Ef(,1' '2' ... ) = Ef('1 + s, '2' 0") = f: Ef('1 + s, '2' .. o)ds =

= Ef: f (,1 + s, '2' .. 0 ) ds = Ef: f (s, '2' .. 0 ) ds =

where , '1

is a random variable in rO,l) which is independent of

with distribution w. Hence 'I has the same properties, and continuing

inductively, it is seen that T l , T2

, are independent with distribution

u). This completes the proof of (ii) when S = [0,1)

Turning to the case of general Sand w with wS = 1 , note first

that the sYmmetry property of ~ extends by monotone convergence from B

to B. For convenience, we may compactify S or rather assume from the

beginning that S is compact. Let S = {S .} c B be a null-array of nestednJ

{I .} formsnJ

S there isS E

I =

For each[0,1]

in such a way that

S .nJ

II ·1 = wS .nJ nJ

S , and note that ~x wSnj+O by compactness of S and diffusenessJ

we can make correspond a subinterval I .nJ

satisfying

To each set

[0,1]

w •

a null-array of nested partitions of

of

of

parti tions of

is measurable,f

Each

s , and hence weS. containingnJ ns

by res) =f: S + [0,1]

a unique decreasing sequence of sets

n 1-.n nJns

t c [0,1] , may be written as a countable union of S-sets, so

can define a function

and it follows in particular that every measure ~ E M induces a measure -1~f

on [0,1]

M , so E,

Moreover, the mapping

induces a random measure

is measurable by definition of

[0,1] .

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Writing D for the set of I-interval endpoints, we get by construction

of f

and hence for any t E D\{l}

-1t ::; wf [0 ,1] tED ,

-1 -1t / wf [O,t] = inf{wf [O,s),! ~ D, s > t} ~ inf{s E D: s > t} 3 t ,

which yields -1 Since is dense in [0,1] this togetherwf [0, t] = t D ,-1 implies that -1

with the fact that wf rO, 1] '" wS = 1 wf equals Lebesgue

measure. Moreover, ~f-1 is symmetrically distributed with respect to

-1wf since ~ is symmetrically distributed with respect to w.

Using the completeness of S, it is easily seen that f has a unique

inverse -1f on f(S)\D , and since and therefore

a.s., there is a.s. a unique correspondence between the atoms of ~ and

-1~f , and (10) is equivalent to

of the theorem is known to be true for

with a.s. diffuse ~ f- I and distinctd

f(T l ),f(T2), ... Now the statement

-1 -1-1~f , and so ~df, = awf a.s.

with a = ~dS , while f(T 1),f(T2), .. , are mutually independent and

-1independent of (a,6 1,6

2, ... ) with common distribution wf . For

every S . E SnJ

we obtain a.s.

~dS .nJ-1

= t;d(S . \f D)nJ-1= t;df (1 . \D)nJ

-1= awf (1. \D)nJ-1

= awf 1 . =nJ

= a I1 . I = awS .nJ nJ

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and S being a OC-semi-ring, it follows by Oynkin's theorem that ~d = aw

a.s. Similarly, we have a.s. for any n, j and k

{'k E S .} ~ {'k € S .\f-lO} = {f('k) € I .\O} = {f('k) E I .} ,nJ nJ nJ nJ

so we get

k kP({(a, Sl' SZ' ... ) E A} (') h. E B.}) = P{(a, Sl' SZ' ... ) E A} IT wB.

j=l J J j=l J

for any measurable set A and for arbitrary kEN and. Bl ,

and by Oynkin's theorem, this extends to general Bl , The

proof of (ii) is now complete. The measurability of (a,S) is obvious.

We now consider the case when wS = <Xl Choose Cl , CZ' ... E B

with CO t S and wC l > a Since Cn~ is synnnetrically distributedn

with respect to C w for each n E N , we know thatn

Cn~ = a (C w) + \ S 0n n ~ nj T .J nJ

for some an' Snl' SnZ' ... and some 'nl' 'nZ' ... , the latter being

independent of (a , S l' S Z' ... )n n n and mutually independent with common

distribution C w/wCn n

Now C ~ = C (C~) for m ~ n , so writingm m n

a = 0. 1 ' Sn I 0 Pmn = wC /wC m $ n E N it is seen by identificationj Snj mn

that 0. 1 = aZ = ... = a a.s. while S is a p -thinning of Sn form mn

m :::' n E N Since Pmn + a as n + <Xl for each fixed m E N , it follows

by Corollary 8.5 that Sm is distributed as a Cox process on R' directed+

and that d Putting A A/weI it is seen thathy some A , A = P A = ,m m mnn

we may take A = (wC ) A for each n Let f E: F be arbitrary, andn n c

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choose n E N so large that the support of f

Then

is contained in Cn

Ee-(~-aw)f = \Eexp{- L S .f(T .)} =j nJ nJ

EE[exp{- L S .f(T .)} I {S .}] =j nJ nJ nJ

= E ITJ. E[exp{-SnJ·f(TnJ·)} I SnJ'] = E IT f exp(-s .f(s))w(ds)/wC •

j S nJ n

and if we define

J-xf (s)hex) = - log e w(ds)/wC

S n

this expression may be written

E IT exp{-h(S .)}= Eexp{- Lh(S .)}. nJ . nJJ J

-S h= Ee n

-S h= EE[e n I 1<] =

= Eexp{-(wC ) f [1 - f e-xf(s)w(ds)/wC ]I«dx)} =n R' S n

+

= Eexp{ - f (1R'xS

+'

-xf(s))e (I< x w)(dxds)} .

Putting f(s) = tIc (s) •1

monotone convergence that

a.s. This proves that ~

t E R • it follows by Fubini's theorem and+

lim L~C (t) = 1 can only be true if I<K < 00

t+O 1is distributed as the random measure described

in (i). But for the in (i) we easily obtain S /wC X I< a.s. by then n

strong law of large numbers, so I< is a measurable function of ~ For

the ~ under consideration we may therefore redefine I< according to

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this mapping, and doing so, (i) will automatically be satisfied. This

completes the proof of the theorem.

Of particular interest is the class of random measures which are

at the same time sYmmetrically distributed and infinitely divisible.

Here is a simple characterization of this class:

THEOREM 9.5. Let the random measure ~ be symmetrically distributed

with respect to some W E Md with canonical random elements (a,A)

or (a, B) Then ~ is infinitely divisible iff this is true for (a,A)

or (a,B) respectively.

PROOF. First suppose that wS < 00 , and let (a,S) be infinitely divisible.

For each n E N we may choose independent random elements

such that

By randomization we may then construct independent sYmmetrically distributed

random measures ~ ~ with canonical random elements'" l' ... , "'n

distributed with canonical random element

(a , B ) •n n

From Theorem 9.4 it is seen that + t;,n

is symmetrically

d... + B ) =

nd= (a, B) , and s 0 ~1 +

d+ t;, = F;.

nSince n was arbitrary, this proves

that F;. is infinitely divisible.

Conversely, suppose that F;. is infinitely divisible and choose for

fixed n E N some independent random measures d dF;.l = ... = F;.n such that

F;.l + ••• + t;,n

d= F;. . By considering the corresponding L-transforms, it

is easily seen that ~l' ... ,F;.n are symmetrically distributed, say with

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canonical random elements (aI' 81), ... ,(an' Sn) . Again we may conclude

dfrom Theorem 9.4 that ~l + . "+~n = ~ has the canonical random element

so we must have+ S )n

+ S )nd= (a, S) Furthermore, the random elements

(aI' 61), ... ,(an' Sn) , being measurable functions of the independent

random measures sl' ... ,sn respectively, are themselves independent.

Since n was arbitrary, this proves that (a,S) is infinitely divisible.

In the case wS = 00 , let us first show that, if sl and s2 are

independent and symmetrically distributed with canonical random elements

(aI' AI) and (a 2 ' A2) respectively, then sl + s2 has canonical random

elements (a l + a2, Al + A2) To see this, note that (aI' AI) and

(a 2 ' A2

) are measurable functions of 1;1 and ~2 respectively, and

hence that 1;1 and 1;2 are conditionally independent with canonical

elements (aI' AI) and (a2 ' A2) , given (aI' a 2 , AI' A2

) Using (9),

it follows that sl + s2 has conditionally the canonical elements

(a l + a 2 , Al + A2) ,given (aI' a 2 , AI' A2) , and the assertion follows

by taking conditional expectations with respect to (a l + a2 , Al + A2)

Once this fact is established, we may proceed exactly as in the case

wS < 00 The proof is thus complete.

According to the last theorem, a symmetrically distributed random

measure t; is infinitely divisible iff this is true for the corresponding

pair of canonical random elements. In view of Theorem 6.5., the distribution

of ~ may thus be described in two different ways by means of spectral

representations of the form (6.6), and we may ask for the relationship

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between the corresponding canonical measures. More precisely, it follows

from Theorem 6.5 that, if ~, (a,A) or (a,S) is infinitely divisible,

then

(11 ) f E F($) ,

or

t E R+

f E F(R') ,+

(13)

respectively, where in (11) rEM while A is a suitable measure on

AO

E M(R:) while y is a measure onM\{O} , in (12) a o E R+

M(R:)\{O} , and in (13) is a measure on N (R') \ {O}+

and we may ask for the relationship between these quantities. For a simple

description of r and A in terms of (ao ' AO' y) or (aD, y) , let

P and Q be the distributions of symmetrically distributed randomX,lJ X,lJ

measures on S with non-random canonical elements (x,lJ) in the cases

wS = 00 and wS = 1 respectively. In the former case we further denote

by Rx the measure on M\{O} induced by w via the mapping s -+ xc ,s

S E S , where x > 0 is arbitrary but fixed.

THEOREM 9.6. Let ~ be symmetricaZly distributed with respect to W E MduJiLh canonicaZ random elements (a,A) or (a,S) . Further suppose that

-130-

~ &s infiniteZy divisibZe, and Zet the quantities (r,A) and

(ao

' AO' y) or (aO' y) respectiveZy be defined by (11) - (13). Then

we have in the case wS = 00

(14) r = A = f P y(dxd~) + f R AO(dx) ,x,~ x

whiZe in the case wS = 1

(IS) r = A = J Q y(dxd~).x, ~

By comparison with Theorem 9.4, it is seen that the integrals

J Px,~dY and J Qx,~dY are formed in exactly the same way as the proba­

bility distributions of general symmetrically distributed random measures,

except that the "mixing" measures y need no longer be normalized and

can in fact be infinite. In addition we have in (14) a term J RxdAO

which can not occur in the corresponding representation of distributions,

since the measure R is infinite and can not be normalized.x

PROOF. Let us first consider the case wS = 00

for any f E F

By Theorem 9.4 we get

log E[e-~f I a,A] = awf + J (1 - e-xf(s))(w x A) (dsdx) .

Wri ting

we thus obtain

g(x) = J (1 - e-xf(s))w(ds) x c R+

-131-

-~f -awf-Aglog Ee - - log Ee

and hence by (12)

Now

.-

-xc fe 5 )w(ds) =

giving rise to the second A-term in (14). Further observe that, by (9),

and so

= f dy(x,~) f (1 - e-mf)p (dm) =x,~

= f (1 - e-mf

) Jp (dm)dy(x,~),x,~

giving rise to the first A-term in (14). Finally, the term aOwf is

of the form rf with r = This completes the proof of (14).

Next suppose that wS = 1

f E F

We then get by Theorem 9.4 for any

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Ee-sf = E exp(-awf - r 8.f(T.)) =j J J

EE (exp (-awf - LB. f (T .) I a, 8] =j J J

f -8.f(.. )= Ee -aw II E(e J JIB.] =

j J

-awf -8.£(T.)= Ee exp L log E[e J J I 8.] =

j J

where

-awf (\' )= Ee exp - ~ g(B j ) =

J

-awf-8gEe ,

g(x) = - log Je-xf(s)w(ds)

Using (13), we thus obtain

, X E R+

But from the above calculation it is seen that

II E N(R~) ,

and hence

= J dy(X,ll) J (1 - e-mf)Q (dm) =X,ll

= J (1 - e-mf

) J Q (dm)dy(x,ll).X,ll

Hence (15) holds, and the proof is complete.

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We are now able to characterize the class of measures r and A

which can occur in the representation (11) of infinitely divisible sym-

metrically distributed random measure. Write for brevity K' (x) :: x .

THEOREM 9.7. Let W E Md and rEM, and let A be a measure on M\{O}

Then there exists some infinitely divisible random measure ~ which is

symmetrically distributed with respect to wand satisfies (11) iff

(i) in the case wS = 00 , (14) holds for some aO ~ a , some

1.. 0 E M(R') and some measure Y on (R+ x M(R~)J\{O} satisfying+

(16) A K < 00 Y{).1 K = oo} = a I (1 _ e-X-).IK)y(dxd).l) < 00

0

."

(ii) &n the case wS = 1 (15) holds for some a O ~ 0 and some

measure y on (R+ x (R~))\{O} satisfying

(17) Y{).IK' = oo} = 0

The quantities (aO' 1.. 0 ' y) or (aO' y) are then unique.

PROOF. Suppose that ~ is an infinitely divisible symmetrically distributed

random measure satisfying (11). Then (11) must be the canonical representation

occurring in Theorem 6.5, and therefore r and A are necessarily unique.

But then (14) or (15) holds by Theorem 9.6 for some (aO' 1.. 0 ' y) or

(uO' y) satisfying (12) or (13) respectively. In the case wS = 00 we

get in particular

r

(18) 1 E -t(a+AK) = a t +- og e 0 t ~ 0 ,

and since AK < 00 a.s. by Theorem 9.4, the last two terms on the right-

hand side of (18) must be finite for all t Furthermore, it follows

from (18) by monotone convergence as t + 0+ that 0 = Y{~K = oo} , which

completes the proof of (16). In the case wS = 1 , note that SKI < 00

a.s., and use a similar argument to prove (17). The uniqueness of

(ao' AO' y) or (aO

' y) follows by the uniqueness assertions in Theorems

6.5 and 9.4.

Conversely, let (aO

' AO

' y) be such as described in (i), and define

r and A by (14). Note that A exists according to the second relation

in (16). By Theorem 6.5 and the last relation in (16), there exist some

random variable a ~ 0 and some random measure on R'+ such that

(a,A) is infinitely divisible and satisfies (12). Using monotone con-

vergence as above, it follows from (12) that AK < 00 a.s., and hence

by Theorem 9.4, (a,A) can serve as the canonical random elements of

some sYmmetrically distributed random measure ~ which by Theorem 9.5

has to be infinitely divisible. Hence by Theorem 9.6, r and A are

related to ~ by (11), and therefore have the asserted property. A similar

proof works in the case wS = 1 .

NOTES. Theorem 9.1 is known for t = 00 in which case it was proved,

independently, by Davidson (1974) (in a special case) and Kallenberg (1973a).

The latter paper is also the source for Corollary 9.2, which extends a result

by Nawrotzki (1962)? Furthermore, Corollary 9.3 is a typical resul t from

Kallenberg (1974a), while Theorem 9.4 was proved by BUhlmann (1960) and

Cf. Kerstan, Matthes and Mecke (1974), p. 80.

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by Kallenberg (1973b) and (1974b), (see also Kallenberg (1973c)). Finally,

Theorem 9.5 extends two results for point processes given by Kerstan,

Matthes and Mecke (1974), pages 66-67, while Theorems 9.6 and 9.7 are7

new.

PROBLEMS.

9.1. Let S = {I, ... ,n} , let w be counting measure on S and let

~ be counting measure on a subset of k elements in S chosen

at rando~. Show that ~ is (in the obvious sense) symmetrically

distributed w.r.t. wand still not a mixed sample process (unless

k = 0 or n). Thus the assumption w E Md is essential in Theorem

9.1.

9.2. Let ~ be a sample process on [0,1] with intensity n times

Lebesgue measure. Show that there does not exist any symmetrically

distributed (w.r.t. Lebesgue measure) point process n on some

interval [O,c], c > 1 , such that the restriction of n to

[0,1] is distributed like ~. (Hint: n[O,l] cannot be non-random

unless n = 0 a.s.)

9.3. Extend Corollary 9.2 to general symmetrically distributed random

measures. (Cf. Westcott (1973).)

Prove the

9.4. Let w E Md and suppose that

S ~ O. Show that ~ and n

distributed w.r.t. w

are

is a S-compound of n , where

simultaneously symmetrically

corresponding result for the

random measures defined by (8.11) and (8.12). (Hint: use analytic

continuation. )

9.5. Let

W •

W E Md be fixed and let ~ be symmetrically distributed w.r.t.

Show that p~-l is uniquely determined by p(C~)-l for any

7 Extensions of Theorems 9.5 - 9.6 are proved in Kallenberg: "Infinitelydivisible processes with interchangeable increments and random measuresunder convolution", Tech. flep0Y't, Dept. of Mathematics, Gljteborg Universlty.

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fixed C € B with wC > O. (Hint: use the connection with thinnings

exploited in the proof of Theorem 9.4. Cf. Ka11enberg (1973c).)

9.6. Specialize Theorems 9.5 - 9.7 to the case of point processes.

10. PALM DISTRIBUTIONS.

Let t; be a random measure on S , and define the corresponding

C b II (p~-l)l S M bamp e measure ~ on x y

(1) B € B M € M.

If Et; is a-finite, which holds e.g. when Et; € M , we can define the

corresponding Radon-Nikodym derivatives P M bys

a.e.P Ms

= (pt;-l)l(ds XM) =

(pt;-l) 1 (dsxM)~[t;(ds);t;€M]

Et;(ds) S € S M € M.

More generally, suppose that f € F is such that Eft; is a-finite.

We can then define

P Ms= E[ft;(ds) ; t;€M]

Eft; (ds) S € S a.e. Eft;

without ambiguity, since the two expressions for P M clearly coincides

a.e. Et; on the support of f . We shall call P M the Palm probabilitys

of M w.r.t. s Note that, by definition, P M is 75-measurab1e in ss

for fixed M E M and that P M is almost a probability measure in Ms

for fixed s E S , in the sense that every probability defining property

holds a.s. Et; Just as in the case of conditional probabilities, we

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may choose a version which is actually a probability measure in M (to

be called the Palm distribution) for every fixed s E S. This is possible

by the usual arguments for construction of conditional distributions (cf.

Bauer (1972)), on account of the following fact.

LEMMA 10.1. M and N are Polish spaces in their vague topologies.

PROOF. By Lemma 4.4, it is enough to consider M. Let G be a countable

base in S , and assume without loss that G is a semi-ring.

Approximate each indicator lG' G E G , from below by a sequence

of functions in Fc

be an enumeration of all these

functions for all G E G , and note that, by Dynkin's theorem, each ~ E M

{~n} , so any sequence

is uniquely determined by

In fact, the latter condition implies

must contain a subsequence

Furthermore, we have

N' c N

j EN.~f.J

j EN.for alliff ~ f. -+ ~f.n J J

compactness of

Nil such thatv

~' (n E Nil) But then ~ f. -+ ~ If. (n E Nil)~n -+ some ,n J J

j E N , so we get ~f. - ~ 'f. and hence ~ = ~ , , proving that indeedJ J

vIt is now easily that the function M2

-+ R defined by~n -+ ~ seen p:+

00

p(~, ~') = Lj =1

~, ~' EM,

is a complete metric on M generating the vague topology. To see that

M is separable, let A and B be countable dense subsets of Sand

R: respectively. Then the class of all finite sums of measures bo ,a

a E A, b E B , is clearly countable and dense in M.

-138-

Henceforth, we shall always assume the set functions ps to be proba-

bilities on M (and even on N if ~ is a point process), and for con-

venience of writing, we introduce for every s E S a random measure

~s ' defined on the same probability space as ~ and with distribution

ps

Just as in the case of conditional distributions, the expectations

corresponding to Palm distributions can be computed in two different ways.

More precisely, we have the following result, known as Campbell's theorem.

LEMMA 10.2. Let ~ be a random measure on S with E~ a-finite, and

let f: S x M-+ R be measurable. Then+

Ef(s, ~s) = f f(s, ~ (w))P(dw) = J f(s,~)P (d~) =Q s M s

= Ef(s,~)S(ds)

EHds) S E S a.e. E~.

PROOF. We have to prove that Ef(s, ~s) is B-measurable and satisfies

(2) J Ef(s, ~ )E~(ds) =B s

E J f(s,~)S(ds)B

B E B •

Now this is true by definition of ~s for indicators f of product sets

A x B, A E B, M EM, and it easily extends by Dynkin's theorem to

indicators of arbitrary sets in B x M. But then it holds for simple

functions, and finally, by monotone convergence, for arbitrary f .

-139-

In the particular case of simple point processes, PF;-l should bes

interpreted as the conditional distribution of F;, given that F;{s} = 1

(Note, however, that a definition along these lines only makes sense when

p{~{s} > o} > 0.) We could then expect F;s to have an atom at s, and

this can in fact be achieved by changing the definition on an EF;-null

set:

LEMMA 10.3. Let F; be a point process on S with EF; a-finite. Then

there exist versions of F;s such that F;s - Os is a point process for

each S E S • For any measurabZe function f: S x N + R we then have+

(3) S E S a.e. E~,

where f(s,~) may be arbitrariZy defined for ~ tN.

PROOF. Clearly

(4) {(s,~) E S x M: ~ - ° E N} = (S x N) n {(s,~) E S x N: ~{s} ~ I} .s

To see that the second set on the right of (4) belongs to B x N , let

B c B be arbitrary and let

of B Then

{B .} c B be a null-array of partitionsnJ

{ (5,11) E B x N: jJ{ s} ~ l} = n u (B . x {jJ EN: jJB . ~ I}) to B x N ,n j nJ nJ

and the measurability in (4) follows from the fact that S is a-compact.

By (2) and (4) we now obtain for any B E B

-140-

f P{~ - 0 E N}E~(ds) = E IBl{~_o EN}(s,~)~(ds) =

B s s s

= E IB1{ i;; {s hll (s ,0 i;; Cds) = E!;B ,

so

f (1 - P{~ - 0 E N})E~(ds) = 0 ,B s s

and hence

(5) p{!; - 0 E N} = 1 a.e. E~.s s

Redefining i;;s == 0 on the exceptional s-set in (5), we can make (5)s

hold for all s E S For each s E S we then redefine ~s = <5 ons

the exceptional w-set where i;;s - 0 4 N to make ~s - <5 E N hold fors s

all sand w. Since ~ B - 8 B remains A-measurable for each s E SS S

and B E B , it is now a point process. But then (3) is an immediate

consequence of Lemma 10.2.

Besides the random measures i;;s we shall also consider, for arbitrary

f E F with E~f E R' the mixture+ '

~f with distribution defined by

M EM.

A

When ~ is a point process, we also consider the point processes f,f

with distributions defined by

P{~f E M} = E~f Is P{!;s 8 E M}f(s)Eqds)5

= E~fEIs l{i;;_o EM}(s,i;;)f(s)~(ds) MEN .5

-141-

It is easily verified that Campbell's theorem remains valid for ~f andA

~f ' in the sense that

(6) Eh(sHfE~f

Here h is any measurable function from M or N respectively to R+

From the definitions of Palm distributions of a given random measure

~ it is obvious that they determine, together with the intensity E~,

the Campbell measure (Ps-l)l and hence also the distribution p~-l .

Indeed, a much stronger result is true, indicating a close relationship

between the Palm distributions for different s E S:

LEMMA 10.4. Let ~ be a random measure on S and let f E F with

E~f E R: . Then Esf and pE,;-l determine the restriction of p~-lf

to the set {~: ~f > O} They further determine p~-l for all s E Ss

a.e. E(fE,;)

Note that f may always be chosen such that f(s) > 0 for all

S E S Then ~f = 0 implies ~ = a , so in this case Esf and

alone determine the whole distribution of ~ .

PROOF. The first assertion follows from (6) by choosing for fixed M E M

~f > 0 ,

~f = 0 ,

-142-

since for this h, (6) takes the form

(8) p{i; E M, i;f > O} = Ei;f Eh(l;f) .

Furthermore, the probabilities in (8) determine the measure

E[ (fS) B; I; E M] E[Cfi;)13; i; E M, H > 0] 13 E B

and so the second assertion follows from the fact that

P{I; E M}s

E[i;(ds) ;I;EM]EI;(ds)

E((fS) Cds) ;i;EM]E (fl;) (ds) S E S a.e. E(fS) .

For f E F , the above uniqueness assertion can essentially bec

strengthened to continuity:

THEOREM 10.5. Let 1;, 1;1' 1;2' be random measures on S and let

condition dthen implies I;nf + I;f ~

Furthermore ~ I; !! I;n

EI; f + El;f ~ then

di;nf + i;fwhi le conversely

dand I;nf + i;f imply that

Under the assumptionEi;f E R~be such thatf ( Fc

implies

EI; f + El;f .n

PROOF. First assume that i; ~ I; and Ei; f + Ei;f , and let h: M+ Rn n +

be hounded and continuous. Then the mapping p -* h(p)pf is continuous,

so we get

(9) h(1; )F, f ~ hCF,)r,fn n

by Theorem 5.1 in Billingsley (1968). Furthermore, it follows from the

relations F, f ~ i;f and Ei; f + Er,f that {E; f} is uniformly integrablen n n

-143-

(cf. Theorem 5.4 in Billingsley (1968)), and hence so is

By (9) we thus obtain

{h(~ H f} .n n

(10)

so by (6)

(11)

Eh(~ )~ f + Eh(~)~f ,n n

is bounded and measurable with

If h:M+R+

Since h was arbitrary, this

Suppose conversely that

dproves that ~nf + ~f

d~nf + ~f and E~nf + E~f

~f ~ Of a.s., it follows by Theorem 5.2

in Billingsley (1968) that (11) holds, and by (6) this implies (10).

For any fixed bounded and continuous function g: S + R we may choose+

__ {]Jo (fg) /]Jfh (]J)

]Jf > a

In fact, h is bounded and measurable, and by (6) the set 0h = {]Jf = O}

satisfies

For this particular h , (10) takes the form

E(f~ )g = E~ (fg) + E~(fg) = E(f~)g ,n n

and since g was arbitrary, it follows by Theorem 4.10 that

as asserted.

wuf E: --)- fE: ,

n

Finally suppose that

-144-

~ i ~ andn

d~nf + ~f ' and choose

h(p) ::: (1 + ).If) -1 . Then h().I) and h().I)).If are both bounded and con-

tinuou5, so (10) and (11) are satisfied. Since moreover Eh(~f) > 0 ,

it follows from (6) that E~ f -+ E~f .n

In case of point processes, we may also prove a continuity theoremA

for ~f' f E Fc

THEOREM 10.6. Let be point processes on S ~ and suppose

that E~ E M\{O}. Then any two of the following statements imply the

third one.

(i)

(li)

(i ii)

Er X. Er<'n <"

A d A

~nf -+ ~f for all f E F with E~f > 0 .c

PROOF. Let f E Fc and let h: N+ R+ be bounded and continuous. Then

the function

(12) g().I) = J h().I - 0 )f(s)).I(ds)S s

).I eN,

is vaguely continuous. In fact, suppose that ).I{s}? 1 and ).I {s } > 1 ,n n

n EN, and that )l '! 11n ,.. in Nand s -+ 5n

in S Then

).I - 0n s

n

holds hy definition of vague convergence, and hence hy continuity

-145-

h (]J - 0 ) f (s ) -+ h (]J - 0 ) f (s ) .n s n s

n

Since hand f has bounded support, it follows by Theorem 5.5 in Billingsley

(1968) that g(]Jn) -+ g(]J). (Note that, in Billingsley's notation, we need

only assume that h (x ) -+ hex) for x -+ x such that x E E and x E En n n n n

= 1 .)

Now suppose that (i) and (ii) hold, and let h and g be as above.

Then Eg(t;) 1 Eg(t.:) follows by uniform integrability as in the preceedingn

proof, and it follows by (i) and (7) that I:h(t;nf) -+ Eh(t;f) provided

Et.;f> o. Since h was arbitrary, this implies (iii).

Next suppose that (ii) and (iii) hold, and define g by (12) with

]J EN.

Then h is bounded and continuous, so by (iii)

(13)A

Eh(snf) -+ Eh(€f) > 0 .

Moreover, g is continuous, and it is even bounded since

l+Jlf-f(s)+llfllf(s)Jl(ds)

IIence by (ii)

(14)

f(s)}l(ds)l+Jlf

Eg (s ) -~ Eg (S) .n

l]Jff <; 1 .+]J

Using (7), (13) and (14), we obtain Et:nf -+ Et;f. If U:f 0, we m~IY

-146-

choose f l , f 2 E Fc with f = f 1 - f 2 and E~fl = E~f2 > 0 , and conclude

that

E~f .

Thus E~ f + E~f is valid for all f E F ,and (i) follows.n c

Finally suppose that (i) and (iii) hold. By (7) we get for any

f E F and any bounded continuous h: N + Rc +

For any fixed

-tffe and put

f E Fc

h(ll) ==

and t E R+ ' we may replace the

-tllfe , thus obtaining

f in (15) by

E I exp{- t~ f + tf(s)}f(s)e-tf(s)~ (ds)S n n

+ E Js

exp{- t~f + tf(s)}f(s)e-tf(s)~(ds) ,

or equivalently

-t~ fE~ fe n + E~fe-t~f .

n

By Fubini's theorem and dominated convergence, we get

-~ f II -t~ fHe n = 1 _ E ~ fe n dt = 1 -

a n

= 1 -

Jl -t~ f II

E( fe n dt + 1 -o n a

I' I;, UC -tcfJt " Eo -(I' ,

EE,fc -tf;fdt

and since f was arbitrary, it follows by Theorcm 4.5 thatdr >- r,

'n

-147-

This completes the proof.

Palm distributions are often easily calculated by means of L-transforms.

In fact, we get by dominated convergence for any f E F with El;f E R'+

(16)

and Ll; is obtained from this derivative by normalization. As a: simples

example, suppose that l; is a Poisson process with intensity A E M\{O} .

By dominated convergence we get

so putting t = 0 ,

and hence by normalization

Lr (f) = Ll;(f)e-f(S) = Ll;(f)L8 (f) = Ll;+8 (f)J s 5 s

S E Sa. e. El;

By Theorem 3.1 thus obtain E;sd

E; 8 equivalently t: s 8d

(we = + or -s s

El; which yields E;fd

E; for f F with Et;f R' Wea.c. , every E E" +

shall usc this fact to prove the following complement to Corollary 7.4.

A ( r~\ {O} , Ze t; I ( BA

of point processes on s

be a DC-serni-ring and Zrt

Put s = L l; .• Thenn . nJ

J

{F, .}nJ

t; ~ t;n

l'e a null-or-rlay

and

-148-

for aU f E F with El;f E R' iff+

ei) L P{I; or > O} -+ AI r E I ,j nJ

(i i) L E[E; 0 B; F,njB > 1] -+ 0 B ( Bj nJ

PROOF. Suppose that (i) and (ii) hold. Then dE, -~ E,n

follows by Corollury

7.4 since

(17) L P{E, 08 > 1} ~ } ~ E[I; .8; I; .8 > 1] -+ 0j nJ J nJ nJ

B E B .

Furthermore, we get by ei), eii) and (17) for any I E I

L EI; .r := L P{I; or> O} - L P{I; or> l} + L E[I; 01; E, or> 1] -+ AI ,j nJ j nJ j nJ j nJ nJ

proving that

:= L EE, 0 X Aj nJ

EE; •

A d A dlienee it fa llows by Theorem 10.6 that E;nf -+ t.f E; for any f ( F with

EE;f E R'+

d A d d A

Suppose conversely that I;n -+ I; and that I;nf -+ S sf for any

f (F with EE;f E R'+

By Corollary 7.4 and Theorem 10.6, we get for

~ P{t; .8 > OJ -+ AB<, nJJ

L P{E; .B > I} -+ 0o nJJ

L U:nj

8,. AB .j

Ilere the fi rst relation contains (i) and moreover

L Efi; .B; ~ .B > lJ =. nJ nJ]

proving (ii).

-149-

L E~ .B - L P{~ .B > O} + I P{~ .B > I} ~ 0 ,j nJ j nJ j nJ

0'

NOTES, Palm probabilities were introduced for stationary point processes

on R hy Palm himself (1943), whose studies were continued by severalv

authors, including llinc'in (1960) and Slivnyak (1962). The present approach

is essentially due to nyll-Nardzewski (1961), (see also .Jagcrs (197:\),

Papangelou (1974), and Kerstan, Matthes and ~1ccke (1974), the latter ;Iuthors

working directly with the Campbell measures). Camphell 's theorem, in the

formulation of Lemma 10.2, was first stated by Kummer and ~Iatthes (1970),

while Lemma 10.3 is implicit in Kallenberg (1973a). The uniqueness Lemma

10,.4 is new. As for the corresponding continuity problem, a discussion

for stationary point processes is essentially given by Kerstan and Matthes

(1965), and for general point processes by Kallenberg (1973a), from which

Theorem 10.6 and Corollary 10.7 are taken. Theorem 10.5 extenns a result

in the same paper.

PROBLEMS.

10. 1. Prove the version of Theorem 5.5 in ~illjngsley (1068) which 1S

used in the proof of Theorem 10.6.

10.2. Let ~ be a random measure on S with Ei; a-finite, and let

M E M Show that P{i; E M} = 1 iff P{i; E M} = 1 , S E Ss

a.e. Ei;

10.3. Let E; be a random measure on S , let B c B with U,B < m

and let

-1 J= x B

x > 0 be fixed. Show that P{i;B x}

p{i; B = x}Ei;(ds) , (cf . .Jagers (1973)).s

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10.4. Let F;, and n be independent random measures on S with T:r,

and En in M Calculate P (F;,-1 in of pF;,-l and+ n) s terms

-1 sPn s

T , S E S ,sSay that a random measure

and define the translation operatorsS = RLet

on M by (T ~)B = ~(B - s) , B c Bs

F, on R is stationa~y if P(TsF;,)-l 1S independent of s E R .

Show that in this case EF;, equals some constant A times Lebesgue-1measure, and that if A E R' , then peT F;,) is independent of s

+ sThe converse is also true. (Cf. Mecke (1967) and Jagersa.e. EF;,

(1973). )

10.5.

10.6. Let ~ be a random measure on R and let f E F (R) with

Show that it is possible to define a random measure ~f on R such

that

f P{T F;, E M}f(s)EF;,(ds)/EF;,fJ -s 5

= E f l{T F;, EM}fF;,(ds)/EF;,f-5 s

~1 EM.

State and prove the corresponding version of Campbell's theorem.

10.7. State and prove a version of Theorem 10.6 for the random measures

F;,f defined in Problem 10.6. Specialize to the case of stationary

random measures. (Cf. Kerstan and Matthes (1965) as well as Kallenberg

(l973a) . )

10.8. Let F;, be a random measure on S and let f (F be such that

EE,f c R'+

Define the di stri hut i on of a random element (n f , r: f )

on R x M by+

for any measurable set A

equals

-151-

in R x M , and show that this probability+

..

State and prove the corresponding version of Campbell's theorem.

11. CHARACTERIZATIONS INVOLVING PALM DISTRIBUTIONS.

Tn the preceding section we saw that any Poisson process [, on

S satisfies the relation ~ - 0 ~ ~, S E S a.e. E~. Actually,s s

this relation characterizes the class of Poisson processes among arbitrary

random measures:

Proposition 11.1. Let ~ be a random measure on S with E[, EM.

Then ~ &s a Poisson process iffd

~ = E. + 8s ' s S E S a.e. Et; .

PROOF.

n ( N

Suppose that d~ ~ + 0 a.e.s s Then we get for any B E Band

n} = !-E{~B; [,Bn n} = !- J E[[,(ds); t;B = n] = !- f P{[, B

n B nBsn}E[,(ds)

=!-J P{[,B = n - l}E~(ds)n 13

and hence by induction

P{~B = n} = P{~B

!- P{~B n - l}E~B ,n

([[,13/O} n!

In the same way we get more generally for any kEN, nl' .. , ,nk E Z+

-152-

and disjoint Bl , ... ,Bk E Bn.

k k k (EI;B. ) J

P n {I;B. = n. } = PH u B. = a} IT J

j =1 J J j =1 J j=l (n.) !J

/\ similar argument also shows that P{~B 1 z } = a ,+

B E B . Hence

must be a Poisson process, and the proof is complete.

The aim of the present section is to prove extensions in various

directions of the ahove proposition. We shall first consider the case

of infinitely divisihle random measures i; with Ei; EM. If a and

A are the corresponding canonical measures defined in Theorem 6.4, we

can define the corresponding Campbell measure A on S x M by the relation

A(B x M) = aBeaM + JM~BA (d~) B E B M E M

Formally, we may write A = ae + Al , where Al =J

pA (dlJ) , d. (10.1) .a

Since J lJBA (d~) = EI;B < 00 we can proceed as in the definition of Palm

distrihutions and define the distributions A s E S a.e. EI; bys

A Ms

= A(dsxM) =A(dsxM)

a(dS)OoM+JM~(dS)A(d~)

a(ds)+fM~(dS)A(dlJ)

Campbell's theorem remains valid for the

negative function f on S

AS

i.e. we have for any non-

A fs

a(ds)f(s,O)+ f(s,lJ)lJ(ds)A(dp)

a(ds)+ lJ(dS)A(d~)

a.e. U:.

-153-

Let us now use the method of the preceding section to calculate ps

For any f E: F t E R+ and B E B we obtain

a~ exp{- af - taB - f (1 - e-~f-t~B)A(d~)}

= L~(f + tlB){aB + J ~Be-~f-t~BA(d~)}

by dominated convergence. Hence

and so, by Campbell's theorem,

L~ (f) = L~ (f)s

Since the last expression is the L-transform of it follows

by Corollary 3.2 that p[-l = p~-l*i\ a. e. , and in particular's sp~-l I p~-l ( p~-l 1S a convolution factor of pt;-l ) . This factorizations s

property characterizes the class of infinitely divisible random measures:

THEOREM 11 .2. Let ~ be a nxndom measure on S z.,)i th H: (M Then

&s infinitely divisible iff s ( S a.e.

a.c.rr- l'5

c(we

1\ much stronger result, to he proved first, holds in the point process

case:

-154-

THEOREM 11 .3. Let E;. be a point procesB on S l,lith EE;. E M 'l'hen

E;. is infinitely divisible iff P{E;.f = O} > 0 and P(E;.f)-l

for every f E F with E;.f ~ 0c

PROOF.

P(E;.f) -1

If E;. is infinitely divisible and f E F with E;.f ~ 0 , thenc

I P(E;.ff)-l follows by the above calculations. Furthermore, Lemma

6.1 yields P{E;.B = O} > 0 B E B , which implies P{E;.f = O} > 0 ,

Conversely, let f E F be fixed withc E;.f ~ 0 and suppose that

f E Fc

Writing ~ = LE;.f ' it follows easily that

~'(t) = ~(t) Ee- ntEF;f t 2> 0 ,

for some random variable n E Z+ ' so if we divide by ~(t) and integrate

from 0 to t , we obt'ain

where

-1T

at + A(l - e t)

a = Pin = O}EF;f; A(dx)-1 -1

x Pn (dx)EF;f x > 0 .

By Lemma 6.1, this proves that E;.f is infinitely divisible. The sufficiency

assertion now follows by Problem 6.5.

Proof of Theorem 11.2. Suppose that IH-- l I -1s Pi';,s

S f: S a .c. ,: F, •

To show that ~ is infinitely divisible, it suffices hy Lemma 6.4 to

pr'ove that U:;BI

, ... ,E,Bk

) IS infinitely divisihle for every k ( N

and every disjoint '\, ... ,1\ ( 13 . Since clearly- 1

Jl r. for

-155-

any B E B with ~B ~ 0 , it is enough to prove the theorem for finite

S , say for S = {l, ... ,k}. For fixed c > 0 and s E S with ~{s} ~ 0 ,

cCE;s{l}, ... '~s{k}) respectively.

and hence also Pn-1 I PCn' + 0 )-1s

Pn,-l

~ = Lc~ , we further obtain

c C~{l},

Writing

, ~{k}) and

From Cl.4) it is seen that Pn- l Ibe Cox processes directed byn'andnlet

from C1. 4)-t -t

~ ~ (l-e 1, ,l-e k)

~ ~ CO, ,0)e-t

s

t 1 , , t k E R+

and so n' + 0S

proving that Since S E S was

arbitrary with ~{s} ~ 0 , we may conclude that PCnf)-l I PCnff)-l for

every f E F , and it follows by Theorem 11.3 thatc n is infinitely

divisible. The factor c > 0 being arbitrary, this implies by Lemma

8.6 that t; itself is infinitely divisible, and the proof is complete.

We are now going to show that it is enough in Theorem 11.2 to consider

one particular mixture of Palm distributions, provided we add an assumption

about the constant part of ~. Given any f E F with t;f < 00 a. s., we

define the non-random set function f~ on B by ft;B = essinf (f~)B =

= inf{x > 0: P{(f~)B < x} > o}. Since ft; is automatically finitely

superadditive and continuous at ~, only finite subadditivity is needed

to ensure n to be a measure. This holds in particular if essinf E;f = 0 .

-156-

THEOREM 11.4. Let ~ be a random measure on S and let f E F be strictly

positive with 0 < E~f < 00

finitely subadditive and

is infinitely divisible iff f~ is

PROOF. Suppose that ~ ~ I(a,A) . Then f~ = fa EM, (this is a simple

special case of Theorem 6.9), and it follows in particular that f~ 1S

subadditive.

Suppose conversely that f~ is subadditive (and hence a measure)

and that p~-l I p~il. Since f is strictly positive by assumption,

it is easily seen that ~ and f~ are simultaneously infinitely divisible,

and so it is enough to assume that f = I = IS' By Lemma 6.4, it suffices

to prove that (~Bl"" ,~Bk) is infinitely divisible for every kEN

,Bk E B. From the assumed factorization property it is

seen that

E[~S exp(-L.t.~B.)]E~~ J J = E exp(- r.t.~B.) E exp(~ L.t.n·)• JJ J JJJ

~

for some R+-valued random variables nl , ... ,nk ' and writing ~

we get

~ - I

(1)E[~S exp(-L.t.~B.)J E~S E exp(-L.t.n.)-~S

J J J Y. ~ J J J -----~......<---"--- = E exp ( - .t. ~B. )

~ 'J J JE~S E~S

Thus the numerator on the right is positive, and letting t l , ... , t k -~ 00

we obtain E~S P{n l = ••• =n k = O} 2 IS. But then the second factor on the

right of (1) must be an L-transform, and (1) shows that the assumed factori-~

zation property of ~ remains true for ~. We may therefore assume from

now on that ~ = 0

-157-

Let us now suppose that 11 = (11 0 , 11 1 , . o. ,11k) is;) Cox procc s s on

{O,l, ... ,k} directed by (I;;S, I;;B I , ,I;;Bk) 0 Proceeding as in the

proof of Theorem 11.2, it is seen that Pll-1 I Pll,-l where Pn,-l denotes

the Palm distribution of n w.r.t. the point 0, and if ~ is the gen-

erating function of n , we obtain

."

(2)t ~ I (t, t 1 ' ... , t k)

Ei;;St, t l , ... ,t

kE [0,1] ,

for some Z+-valued random variables S, sl' ... ,sk ' where the prime stands

for differentiation with respect to t. Now

11HO, t l , '0. ,tk) = E[t l

l

so it follows from (2) that t is a factor of the expectation on the

right, i.e. that P{s = a} = a Rewriting (2) in the form

1/J'(t,t l ,···,tk) 1 sl sk

H t , t 1 ' . . . , t k ) = EI;;S Et s - tl t k

and integrating w.r.t. t from a to 1, we thus obtain

In particular

a - log 1/J(I, ... ,1)-1

- log HO,I, ... ,1) - E~S Es

and hence by subtraction

- log 1/J(1, t l , ,tk) =

log HO,I, ,1) - log 1/J(0, t l , ... ,tk) + I:F;S Es -1

(1 - t:l

Since

-158-

-1Es < 00 , it is seen from Lemma 6.3 that the function e on

[O,l]k defined by

tl

, ... ,tk

E [0,1] ,

1S the generating function of some infinitely divisible random vector

on Zk+

Letting

and writing

for any p E

be a Cox process directed by

~p for its generating function, we obtain more generally

(0,1]

(3) - log ~p(l, t l , ,tk) =

= log ~p(O,l, ,1) - log ~p(O, t l , ... ,tk) - log ~(tl' ... ,tk)

where Gp is an infinitely divisible generating function. But n is

by Corollary 8.5 a p-thinning of n , and so by (1.6) and (3)p

where q = 1 - p. Here the last term corresponds by Lemma 8.6 to an

infinitely divisible distribution, so if we can show that

(4 ) log t/Jp(O,I, ... ,I) - log ~p(O, q + pt l ....• q + ptk) + 0 as p + 0 ,

-159-

it will follow that ~(l, t l , ... ,tk) is the limit of a sequence of

infinitely divisible generating functions, and hence itself infinitely

divisible.

.'

To prove (4), put ¢ = L , and note that by (1.4)~S, ~Bl ' ... , ~Bk

E exp{-p-l(~S(l - t) + L.~B.(l - t.))} =J J J

Hence

(-1 -1 -1)

= cpp (1 - t), P (1 - t 1)' ... ,p (1 - t k )

while

~ (0,1, ... ,1)P

-1 -1= ep(p , 0, ... ,0) = E exp (-p ~S) •

-1~ ~P (0 , q • ... , q) = cp (p ,1, '" ,1 )

= E exp(-p-l~S - L.~B.) ~ E exp(_(p-l + k)~S) .J J

Now it may be seen as in the proof of Theorem 11.3 that d~S = some l(a,>') ,

and since ~ = 0 we have a = O. Thus by Lemma 6.1, the difference in (4)

is at most equal to

log E exp(_p-l~S) - log E exp(_(8P-l + k)~S) =

which tends to zero as p + 0 by dominated convergence. This proves

that the Cox process (n l , ... ,nk) directed by (F,31 , ..• ,£:1\) IS

infinitely divisihle, and exactly the same argument shows more generally

-160-

that Cox processes directed by c(~B1'

for any c > o. By Lemma 8.6,

. .. , ~Bk)

(~B1' ...

are infinitely divisible

,~Bk) is then itself

infinitely divisible, and the proof is complete.

We shall now prove a related but quite different criterion for infinite

divisibility which applies to the class of purely atomic random measures

with diffuse intensity. Given any measure ~ E M and any E > 0 , we

define the measure ~ E M byE

~ B =E

B E B ,

and note that ~E is discrete in the sense that it is purely atomic with

finitely many atoms in each compact set. Note also that the mapping

~ ~ ~E is measurable by Lemma 2.1. Given any random measure ~ on

S , and any B E Band E > 0 with P{~ B = o} > 0 , we further defineE

the random measure ~B,E on S as one with distribution

P{~B E M} = P{~ E M I ~ B = o},E E

M EM.

THEOREM 11.5. Let ~ be a purely atomic random measure on S with

Md Then is infinitely divisible iff-1 I p~-l for everyE~ E ~ PB,E

E > 0 and B E B 7,n th P{~ B = o} > 0 . In the case 7'Jhen ~ &s a. r,.E

discrete~ this statement remains true w1:th E = 0

PROOF. The necessity is an immediate consequence of Lemma 6.6 and the

fact that Poisson processes have independent increments. To prove the

-161-

is independent of nB . For

for every B E: B satisfying P{~B = O} > 0

that ~d

~B where ~ ~ ~B 0assume + nB B ,s ( B a.e. E~ and any f E: F we obtainc

-1 I -1converse, let us first assume that ~ is a.s. discrete and that P~B,O P~

For any such B we may

L[ (f)"'5

(5)

and this holds simultaneously for all B belonging to some countable

base G and all $ E: B except for points s E: S in some fixed E~-null

set C. For every fixed s 4 C , consider a sequence Bl ,B2 , .. , E: G

such that Bn + {s} Since ~ is a.s. discrete and has no fixed atoms,

have P{~B O} 1 and ~Bd

~ Hence by (5) and Lemma 5.1,we -+ , so -+n

d p~-ln

nB s -+ some n , and we get p~-l We may thus conclude fromsn

Theorem 11. 2 that ~ is infinitely divisible.

We now consider the general case al1d hence assume that p~;~ I pt;-l

whenever p{~ B = O} > 0E

Writing d~ = t;BE + n BE with independent terms,

we get in place of (5)

and so

< 2

-162-

IL~ (f) - Lt,;(f)Ln (f) Is BES

Et,;BE(ds)E~(ds) + ILt,;BE (f) - L~(f) I

-t,;fE[~(ds);t,;cB=O] E[e ;t,;cB=O]

2 ~ ~ E -~f= =E-t,;(7'd-:--s-=-)=p"""{t,;=--=B-=-=-O'"} + ---'P"""{'-t,;---=-B=-'O"'""'}-- - .e

E E

E(t,;-t,;E) (ds) 1

2 ---=E=-:"t,;-=(-=-ds"""""')-- P{t,; B= O} + 2E

l-P{t,; B=O}E

P{t,; B=O}E

As before, this holds for all s outside some E~-null set C, simultaneously

for all E in some sequence E tending to 0 and all B with P{t,; B=O} > 0E

belonging to some countable base G. Now ~ - ~E X0 a.s. as E ~ 0 , so

E(~ - t,;E) X0 by dominated convergence. Choosing g (s) :: E(t,; - ~ ) (ds)/E~(ds)E E

non-decreasing in E E E , it follows from this by dominated convergence that

g (s) ~ 0 as E + 0 for s E S a.e. Et,;, say for s outside some null­£:

set C' Furthermore, every ~E is clearly a.s. discrete and has no

fixed atoms, so for fixed E we have P{t,; B = O} ~ 0 asE

B -} {s}

Given arbi trary n €. Nand s 4 C u C' , we may therefore choose f. ( E

so small that g (s) < (8n)-1 and then BEG so small thatE

-1p{[ B > O} < (8n) . Writing nn for the corresponding random measure-E

nBES ' we then obtain

-1< n n E N f <: r

c

proving that By Lemma 5.1 it follows that some

-163-

. f' L L L d' p~-l I p~-l . A b f h" l'satls ylng t;, n = t;, , an agaln s Ss s e ore, t lS lmp less

infinite divisibility of t;" and the proof is complete.

We are now going to consider extensions of Proposition 11.1 in another

direction.

THEOREM 11.6. Let ~ be a point process on S lJith Et;, E: M Then

P(~s6 )-1 independent of S Et;, iff t;, d

M(Et;" tjl) fOY'- 1,fj S E: a.e.s

tjl Tn this tjl'(O) -1 and t;,s 6d

M(H, -tjl')some . case = -' - a.e.s

It is interesting to see how the specialization of Proposition 11.1

to point processes follows from this result simply by solving the differential

equation tjl = -tjl' subject to the initial condition tjl(O) = 1

The proof of Theorem 11.6 is based on the following lemma, giving

an interpretation of the notion of Palm distributions in terms of ordinary

conditional distributions.

LEr~MA 11. 7. Let ~ be a p01:nt process on S with EE; E: M -' and let

n I n , C E. B and M E: N For ~C > 0 u)C fuy,theY' asswne that T

1AJ one of ihe atoms of E, &n C -' chosen at random. Then

p{~ E. M I ~C = n, T = s}

a. e. u)i th Y'eBpect to the measure

P{t;, E: MIt;, cs sn} s ( C ,

E[E:(ds); ~C nJ = P{ E; Cs

n}EE;(ds) s c S .

PROOF. The n atoms in C being symmetric, we get for see

-164-

P{E; E M, E;C n, T E ds} = E[o (ds); t;, E M, E;C = n]T

-1t;,C = n]= n E[t;,(ds); t;, E M,

= n-lp{t;, E M, t;, C = n}Et;,(ds)s s

and in particular

P{t;,C = n, T E ds}

By the chain rule for Radon-Nikodym derivatives, we thus obtain a.e.

p{E; E M I E;C = n, T = s} = P{t;,EM, E;C=n, TEds}P{ E;C=n, TEds}

P{ t;, EM, t;, C=n}s sP{ t;, C=n}

s

as desired.

= P{E; E MIt;, C = n} ,s s

Proof of Theorem 11.6. Suppose that dt;, = M(Et;" ¢) . As shown in Section

9, E;B has then the generating function ¢(EE;B(l - t)) , t E [0,1] ,

and we get in particular

P{E;B = l} = d~ ¢(EE;B(l - t)) I t = ° = - EE;B ¢' (EE;B) B E B .

Using this fact and Corollary 9.2, we obtian for any B,C E B with B c C

f P{ (E;. - 0 ) B = o}EE; (ds )r: s s

= f P{E; Br: S

l}EE;(ds) = qr,C; r,B 1]

= P{E;B = l}E[~C I eB

- cf>'(EE;B)Ef,C .

1] _. 1;[,13 ,j,' ((;[,13) H.C'I' 1:F;B

On the other hand we get hy Corollary 9.2 for any disjoint B,C E B

-165-

I P{(I: -8)B O}EI:(ds) = Ic P{I: B O}Eqds) E[I:C; I:B 0]C S 5 5

00

I nP{I:C n, E;B = O}n=l

00

= I np{qB u C) = n} P{ E;C = n I E;(B u C) n}n=l

00

{ HC r= I nP{t;(B u C) n} EE; (BUC)n=l

00

EE;C d \' nEqBuC) dt L. t P{E;(B u C) = n} I

n=O t EE;Cjl:[, (sue)

EE;C d ( ) I= EqBuC) dt <PEE; (B u C) (1 - t)t = EE;CjEE;(BuC)

EEC ( )EE;(BuC) EE;(B u C) <P'EE;(B u C) - EE;C

- - <P' (EEB) EE;C

These calculations show that for any B E B

P{(~ - 8 )B = O} = - <P' (EEB)5 5

5 E S a.e. EE;.

If EE; c Md ' then

by Theorem 3.3 that

to evaluate

[, and hence also E; - 8 is simple, an<1it followss s

E - 8 ~ M(EE; -<P') Tn the general case we have5 5 ,.

n l' ... , (E; - 8 ) Bks 5keN

But these probabilities arc determined hy the expectations

-166-

E[s(ds); sBl = n l , ... ,sBk = nk] , so they are uniquely determined by

the finite-dimensional distributions of s which in turn depend on Es

and ~ only. Thus it makes no difference whether Es E Md or not, and

the above conclusion for diffuse Es must be generally valid.

Conversely, suppose that c - 0"'s s

d some n a.e. Es . Let C E B

and n c N be fixed with n} > 0 , let ,B E Bm be an

arbitrary disjoint partition of C and let k l , ... , k E Zm +

T as in Lemma 11.7, we get for s E Bl a.e. E[E;(ds); sC := n]

P{(s - 0,) Bl:= kl , ... , (s - o ) B := k I sC := n, , :: s}, m m

Defining

, sB := k I sC := n, , = s}m m

= P{ ssBl:= k + 1, ssB2

:= k2 , ... , ssBm:= k I s C := n}

1 m s

= P{(s os) Bl:= k l , ... , (ss - o )B := k I (ss - o )C := n - l}

s s m m s

= P{nB l:= kl , , nB = k I nC := n - l}

m m ,

and by the symmetry in Bl , ... ,Bm ' this remains true for SEC a.e.

E[E;(ds); E;C := n] . By Dynkin's theorem, we can extend this result to

P{E; - 0 E M I sC = n, ,, s} := P{n E M I nC n - l}

proving that , and s - 0 are conditionally independent, given that,

in C , we may easily conclude that

is a random cnumeration of the atom positionsr,C = n . If Tl

," n

,.J

is conditionally independent of

h.,i~j}]

for every j E {l, ... ,n} But then must

Ilc conditionally independent, and since they arc also equally distributcd,

-167-

Ci; is conditionally a sample process. Furthermore, we have for any

B E B with B c C

oW

-lE[eB I eC = ] = E[i;B; i;C=n]n ,s s ,n n P{i;C =n JPh 1 ( B I i;C n} E[o B I i;C = n]

'1

IBP{i;sc=nJEi;(dS) = ]BP{(i;s-os)c=n-1H:i;(dS)

n P{i;C=n} n P{i;C=n}

=P{nC=n-l}Ei;B

n P{i;C=n}

= Ei;Bfcp{(i;s-os)c=n-l}Ei;(dS)

n Ei;C P{i;C=n}

= Ei;B E[SC; i;C=n]n Ei;C P{i;C=n}

Ei;Bn Ei;C E[i;C; i;C=n] =

Ei;BEi;C '

showing that the common conditional distribution of the ,.J

is independent

of n. It follows that Ci; is unconditionally a mixed sample process,

and an appl ication of Coroll ary 9.2 completes the proof.

We shall finally consider an extension of Theorem 11.6 to gener;l1

random measures. I{ecall that a measure is deycner'ate if its mass

IS confined to one single point. If n is a random variahle while 1';

is a random measure on S , then (n,s) is said to be symmetrically

distributed w.r.t. some w EM, if the distrihution of (n, sBl'

for arbitrary kEN and disjoint Bl

, ... ,Bk E B only depends on

For convenience, the random measures descri hed in parts

(i) and (jj) of Theorem 9.4 arc said to he of Type 1\ and B respectively.

We shall a 11 ow the measure w there to he fi n i te in both cases. Note

that the two classes of random measures will then no longer he mutually

exclusive. The canonical random elements (lX,/-) and (a,()) :lrc tlefined

as in Theorem 9.4, and in case of Type 1\ random measures, we further

-168-

define the canonical random measure A on R+ by A(dx) = aoO(dx) + x\(dx) ,

x z 0 .

THEOREM 11.8. Let ~ be a random measure on S ~ith E~ E M Then

d~ - ~ {s}o ) = some (n,s) independently of s a.e. EF, iffs s s

(except possibly for a.s. degenerate ~) and ~ is symmetrically

distributed ~.r.t. E~ In this case, (n,s) is also symmetrically

distributed ~.r.t. E~, and furthermore, n is independent of s iff

either

(i)

(ii)

~ 'l-S of Type A ~ith A = pM a.s. for some random vaY'1:ab le

p ~ 0 and some non-random M E ~~ (R ) , or+

F, is of Type B ~ith a = 0 and S a mixed sample process

on R'+

Note by comparison with (i) that the point process S in (ii) is

also allowed to be a mixed Poisson process. In this case we can even

take a non-zero, proportional to the mixing variable.

To save space, we omit the rather complicated proof of this theorem

and refer the reader to Kallenherg (1974b).

NOTES. Proposition 11.1 was proved for point processes by Kerstan and

Matthes (1964) and Mecke (1967), but the prescnt proof is adapted from

.lagers (1973). Theorem 11.2 is clue to Kerstan and Matthes (1~)64), Mecke

(1~)67), Kummer ancl Matthes (1970) and others (cf. Kerstan, ~1atthes and

Mecke (1974), page 116), while the extensions given in Theorems 11.3 and

-169-

11.4 are new. Theorem 11.5 extends a result for point processes, proved

by Matthes (1969) with entirely different methods, (cf. Kerstan, Matthes

and Mecke (1974), page 97). Finally, Theorem 11.6 is due to Slivnyak

(1962), Papangelou (1974) and Kallenberg (1973a), while Theorem 11.8 was

given in Kallenberg (1974 b).

PROBLEMS.

11.1. Let r; be a point process on S with Et; E: M and let I c 13

be a DC-semiring. Show that t; is infinitely divisible iff

kfor every function f = I II ' k E: N ,

j =1 j

II' ... ,Ik E: I , such thatdt;f ~ 0 •

11.2. Suppose that S is countable. Show that there exists some fixed

such that, given any point processfunction

is infinitely divisible iffwith

and

f E: F

t;S < 00 a.s., t;

P(t;f) -1 I P(t;ff)-l (Hint:

E, on S

P{t; = O} > 0

use the result in Problem 6.4.)

11.3. Let t; be a point process on S and let B E: 13 be such that

Pit; ~ 0, E,B = a} = O. Show that E, is infinitely divisible iff

Pit; O} > a and pt;-l I pr;;l f = lB (Hint: argue with r;

as with the Cox processes in the proof of Theorem 11.4, and note

that the counterpart of (4) holds trivially in the present case.)

11.4. Show that the condition Er;; E: Md in Theorem 11.5 can he replaced

hy the assumption that EF, be a-finite and diffuse.

-170-

a.s., and let

~ ~ 0 has the con-

11.5. Let ~ be a random measure on

l be a random element in S

ditional distribution ~/~S

measurable f: M x S -)- R ,+

S with ~S < 00

which for given

Show that, for u > 0 S E S and

E[f(~,T) I ~S = U, T = s] = E[f(~s' s) I ~sS = u]

holds a.e. E[~(ds); ~S E du] , (cf. Kallenberg (1974b)).

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Abe1schen Gruppen. 7.. WahNlcheinUchke1:tsthcoY'ic ver'l,). Gehiete.

9.36-58.

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stochast ischen Poi ssonschen prozcsse. /'. h'ohr',:dz('{nliehkc1' L,:f)W()f'ir'

()(~ r'lJ). (;elJ1: (::I;e . 11 . 7'1 - R] .

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W'iumen. Be1:tr'?t(!e ;;W' Analysis. 3. 7-:)0.

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tion, Technischc Ilochschule Karl-Marx-Stadt, Section Mathemat iL.

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(46) Manch, G. (1971). Vera11gemeinerung eines Satzes von A. R~nyi.

Studia Sci. Math. Hungar. 6. 81-90.

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Punktfo1gen.

(1962). Ein Grenzwertsatz fUr homogene zuf~llige

Math. Nachr. 24. LOl-217.

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539-541.

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-175-

AUTHOR INDEX..

Bauer 4 9 41 45 137

Be1yaev 29 111

Bi 11 ings ley 4 40 43 44 47 51 52 54 55 62 65 72 91100 101 104 109 142 143 145 149

BUh1mann 134

Daley 29

Davidson 134

Feller 86 116

Fomin 6

Gihman 29

Goldman 98 III

Grandell 37 55

Grige lioni s 98

Harris 37 55 65

Hin~in 98 149

Jagers 55 86 98 149 150 168

Kallenberg 1 17 29 37 55 56 67 86 98 III 134 135136 149 150 168 169 170

Karbe 65

Kerstan 1 17 38 65 86 III 134 135 149 150 168 169

Kingman 98

Ko1mogorov 6

Kummer 37 111 149 168

Kurtz 55 65

Leadbetter 29

Lee 86

Lo~vc 12 38 62 73

Lukacs 87 112

M..'ltthes 1 17 37 38 65 86 111 134 135 149 150 16R 169

Mccke 1 17 37 38 65 86 III 134 ns 149 ISO 168 16~)

Monch 37

Nawrotzki III 134

Ososkov 98

-176-

Author Index (cont.)

Palm 98 149

Papange10u 149 169

Prohorov 37 SS 65

Renyi 37 III

Ryll-Nardzewski 149

Sierpinski 9

Simmons 63 85 138

Skorohod 29

Slivnyak 149 169

Wa1denfels 37 S5

Westcott 13S

-177­

DEFINITION INDEX

bounded set

Campbell measure

Campbell's theorem

canonical rantlom clements

completely monotone

compound

covering

Cox process

DC-ring, DC-semiring

degenerate

diffuse

discrete

tlistribution

Dynkin's theorem

intlepcntlent increments

infinitely divisible

intensity

L-· transform

mixed Poisson process

mixed sample process

mixing

nested partitions

null-array of partitions

null-array of random measures

Palm distribution

point process

Poisson process

random measure

rcI!, lda r

sample process

simple point process

symllletrically distributed

6

136 152

138

120

63

17

7

16

6

87

18

160

11

9

87

67

11

11

16

14

12

2S

20

66

137

10

16, 17

10

24, 28, 48, 92, 93

14

28

113, 167

-179-

SYMBOL INDEX ,

p 6 aB 26

S 6 1;* 28

B 6 ].1 V 30

B 6d

31=F=F(S) 6 II . II 32

F =F (S) 6 d39-+

C C

M=M(S) 7 BI; 40

N=N(S) 7 ~, ~ 41

Z 7 ~ 42+wd].1f 7 ~ 52

M 7 t!.A 63

N 7 1Tt

,1Tf

,1TV

67

BC7 K 67

--B 7 lims i 68

f].l 8 R' 68+

Bp 8 E 68£

113

8 I(a,A),I(A) 78

0(' ) 9, 18 M(w,<P) 114

(rt,A,P) 10 P Q R 129-1 X,].1' X,].1' x

PI; 11 P 136s

EE.: 11 !;s 138~

I.e 11 !;f,l;f 140

0 14 f\ 152s st~J 18 I 153

Po 18 ft; 155

\{' 19 lJE: 160+

Z 19 E, IbO+ 13,£

I•I 20

11* 20,1J' 20

f

r k '"J h) 22" "

-180-

INDEX OF SOME SCATTERED TOPICS

atomic properties

canonical representations

compound point processes

convergence in distribution

Cox processes

independent increments

infinite divisibility

mixed Poisson and sample proc.

null-arrays of random measures

Palm distributions

Poisson processes

sample processes

simplicity and diffuseness

symmetric distributions

tightness criteria

thinned point processes

uniqueness

18ff, 56, 80, 99

68ff, 89, 120, 133ff

17, 32, 36, 99ff, 112, 135

39ff, 57f, 67ff, 87ff, 99ff, 142ff

16, 30, 32, 38, 56, 105, 109ff, 125, 157f

87ff, 105, 119

67ff, 87ff, 112, 127ff, 152ff

14, 16, 113ff, 163, 168

66ff, 89ff, 147

136ff, 151ff

16f, 78, 91f, 118,147,151

14, 118, 167

28, 32ff, 48ff, 58, 64, 93ff, 114ff

113ff, 167f

44, 70

17, 30, 34, 56, 100, 109ff, 112, 125, 136, 158

30ff, 93, 105, 141