A-AIII'"7 STANW W4IV CA ItiS? FOR MATtEMATICAL'STUDIES IN TM--ETC F/e 12/1INFINITE IXCESSIVE AND INVARIANT MEAILPRES. (UlMAY @I M I TARSAR mocoOIN79-c-o0sS

WICLASSIFIED TR-334 Ft.

ummmhhhhE..hMMOMMOOEEHE

11111 1.0$1.2 2

4 11111L25

:- j.6IHI1.251 IIfII JJIII1.6

MICROCOPY RESOLUTION TEST CHART

NAIJONAL BALIA (A IANIIARI I 4 A

I.;.INFINITE EXCESSIVE AND INVARIANT MEASURES

by

Michael I. Taksar

. 00

Technical Report No. 334

p May 1981

A REPORT OF THE

CENTER FOR RESEARCH ON ORGANIZATIONAL EFFICIENCYSTANFORD UNIVERSITY

Contract ONR-NO0014-79-C-0685, United States Office of Naval Research

THE ECONOMICS SERIES

INSTITUTE FOR MATHEMATICAL STUDIES IN THE SOCIAL SCIENCESFourth Floor, Encina Hall

Stanford University AStanford, California ... "

94305 W

INFINITE EXCESSIVE AND INVARIANT MEASURES*

by

Michael I. Taksar

1. Formulation of Results J

1.1. "in the paper [9 the following problem was considered. Given

a contraction semigroup Tt on a Borel space D a d an excessive measure

v, when is it possible to find another contraction se igroup T t such

that t -> Tt and v is invariant with respect to T t. The most restrictive

condition under which this problem was solved is the finiteness of the

excessive measure v. This condition excludes such an interesting case

as the semigroup Tt generated by the transition function of Wiener's

process and the Lebesque measure v. In the present paper we extend the

results o f.[9] to all quasi-finite null-excessive measures v.

Definition. Let Tt be a semigroup. A measure v is called null-

excessive with respect to Tt if for each r C D, subject to v(r) <

vTt(r) + 0 as t-

An excessive measure v is called quasi-finite with respect to T

if for some s > 0 the difference between v and vT is a finite measure.

The principal part of the proof of the main res, the same as that

of [9]. We consider the transition function p which generate Tt, then we con-

struct a stationary Markov process (w(s),P) with the transition function p

and the one-dimensional distribution v. (Actually the process w(.) has random

*This research was supported by the Office of Naval Research Grant

ONR-N0014-7-C-0685 at the Center for Research on Organizational Efficiency,Stanford University.

" " . . . . . .. . . ...u . . . ' - , - :i , n n m ,/, ".. .. . . . . " . . . . , _ . ; :.. ,,, _ L u -.. ,4.,. -. . .

-2-

birth and death times and the measure P is infinite.) We add a single point V

to the space D and we look for a stationary Markov process (xt,P) with the

state space E D U V such that

l.l.a. The birth time of x t is equal to and the death time

of xt is equal to +-.

l.l.a. The one-dimensional distribution of P is equal to v.

l.l.y. p(t,x,r) = P xXt E r; xs E V for all s < t)

A process (xt,P) satisfying l.l.c-l.l.y is called a covering

process for (w(s),P) (see [8] for a more detailed discussion). If the

measure v is infinite then so has to be P, and we cannot apply the

results of 18] for the construction of (xt,P). In order to extend the

results of [9] to infinite measures v we have to develop the whole theory

anew. Accordingly, all the definitions and notations of [7] and [81 will

be used without explicit mentioning.

In the second part of this section we give precise formulations

of the main results and give the conditions under which they are proved.

In Section 2 we prove the existence of (O,R)-generated random set M

for any measure R which is the Levi measure of an increasing process

with independent increments. (In [71 such sets were constructed only for

R having the first moment.) Here the most important tool is the theorem

of B. Maisonneuve in [61, which enables us to find an invariant distribu-

tion for the "Jump process" of the process with independent increments.

Using this result, we prove the existence of a covering process for any

stationary Markov process with a quasi-finite one-dimensional distribution

p

(Section 3). Section 4 is devoted to the construction of a semigroup Tt

with respect to which v is invariant.

In the case when the proof is similar to the one given in [71, [8],

or [9], we shall only outline it, without going into details.

As always the same letter is used for a measure and the integral with

respect to this measure. The word "function" stands for a nonnegative bounded

measureable function.

1.2. Let D be a Borel space and Tt, t >- 0, be a semigroup

in the Banach space of bounded measurable functions on D (we say for

brevity that Tt is a semigroup on D). The semigroup Tt is called

a positivity preserving normal contraction semigroup if

1.2.A. For any t 2- 0 and each function g Z 0 N

INSPE TEZT tg 0

1.2.B. For each x E D rg.S

Ttl(x) , and lim Ttl(x)1 --t t

1.2.C. If f(xO ) = 0 then Tof~x ) = 0.0 0 va :/ L---.JI

DIZ esList .. . -: ? :es

A semigroup T is called continuous if 't/.

1.2.D. For each x E D - I I

Ttl(x) is a continuous function of t > 0

A positivity preserving normal contraction semigroup is denoted S-semi-

group. If S-semigroup Tt satisfies 1.2.E below, then Tt is called

-A -. A -

dying or SD-semigroup; if Tt satisfies 1.2.E', then Tt is called con-

servative or SC-semigroup.

1.2.E. For each x E D

lim Ttl(x) = 0

1.2.E'. Ttl E 1 for each t > 0

(Note that 1.2.E' implies 1.2.D).

If Tt and Tt are two semigroups on D and for each function g

(1.2.1) Ttg S Ttg

then we say that Tt is larger than Tt, or Tt is an enhancing of Tt '

We write Tt = Tt a.e.p if for any function g for u-almost

all x Ttg(x) = Ttg(x).

In this paper we are going to prove the following theorems.

Theorem 1. Given a continuous SD-semigroup Tt and a quasi-finite

null-excessive meaaure V, one can find a SC-semigroup Tt which is

larger than Tt and for which v is invariant.

Theorem 2. If Tt and v satisfy the conditions of Theorem 1

and if it, addition v is an extreme excessive measure then Tt is

unique up to the measure v.

2. Regenerative Sets with Infinite Underlying Measures

2.1. Let (,F) be a measurable space and Q be a measure on

F (not necessarily finite). A subset M C T x a is called a random

-5-

set (r.s.) if it is 8 X F-measurable and M(w) is nonempty for a.e. w.

(Here T is the real line ]-m,+w[ and B is its Borel a-field.)

A r.s. M is called closed (closed from the right, perfect, discrete,

etc.) if for a.e. w, M(w) is closed (closed from the right, perfect,

discrete, etc.). Only closed random sets will be considered in the sequeal.

We refer the reader to [7] for the definitions of the associated random

t - tprocess zt, sets Mt, M , Mt, M, etc.; the definitions of regenera-

tivity, translation invariancy, as well as the definitions of (0,I)-

processes, (a,n)-generated set.

A r.s. M is said to have a a-finite distribution (or M is

a of-set) if

2.1.A The process zt has a-finite one-dimensional distributions.

For example, consider any increasing process with independent incre-

ments with the Lebesque initial distribution (i.e. initial distribution

uniform on T). The range of this process is a r.s. whose distribution

is not a-finite. Let us take now any 0-finite measure V with support

on [0,1] and let R be a unit measure concentrated in the point 1.

If we consider the range of the (0,11)-process with initial distribution

v, then this r.s. has a a-finite distribution.

Any measure 11 on ]0,-[ subject to

(2.1.1) fx A 11(dx) <0

may be considered as the Levi measure of an increasing process with inde-

pendent increments (subordinator), and any subordinator has the Levi measure

'L . . _ .- - . .. . . . I : - ., - . ' • . 4 .i. . . ' - "

-6-

satisfying (2.1.1). The range of any subordinator is a right regenerative

set; all translation invariant sets of such type with finite underlying dis-

tributions are described in 171. These sets are in one-to-one correspondence

with the ranges of all (a,R)-processes with R having the first moment.

It is possible to perform a similar analysis for all r.r.t.i. of-sets, but

we restrict our attention only to the theorem of existence.

Theorem 2.1.1. For any a > 0 and any measure n on ]0,-[

subject to (2.1.1) there exists a t.i. (c,n)-generated of-set M. The set

M is left regenerative and moreover, -M has the same distribution as M.

Let the complement of M be the union of disjoint open intervals

]y,S[. Then for any function f on T x T

(2.1.2) Q{Jf(y,6)} = f{ff(ss + y)n(dy))dsY 0oo

2.2 For simplicity of calculations we shall consider only the case of

a = 0. The modification of the proof for a > 0 is trivial. Let yt be a

(0,n)-process and be its transition probabilities. We denote by 0, the

first hitting time of ]I,-[ by yt; and by YP = (URJ,V) (y ,y) we

denote the "jump" process of yt (see [7] Section 2). Vt as well as Yt

is a Markov process. Let

4 %(x) , if x t tq(s,x; tr =)sQxxVt E ) , if x < t , r C T

be the transition function of the process Vt. Let 11(x;-), xW(-), X b etc.

be the measures and the kernels defined in Section 2 of [71. Denote

-7-

Rt = ]-t,t[ Rt S

t1t (r)= fn(x; rdx , C R

By the theorem of Maisonneuve (see (6], Th. (3.2)) the family t is an

entrance law with respect to q. Note that Vt(r) = i0 (r - t). Consider

the Markov process (vtQ) with the one-dimensional distributions

and with the transition function q. (The measure Q is finite iff V

is a finite measure.) The existence of such a Markov process is proved

in [5]. The same way as in Lemma 6.2 of [7], we can show that vt is

a stochastically continuous increasing process; hence there exists a

right-continuous version of it. Consider the random set M which is

the range of v t (i.e. the closure of the set of values of v t). We

are going to prove that M is the set we are looking for.

Lemma 2.2.1. The set M is a translation invariant right-regenera-

tive (0,11)-generated set with the associated process zt having the

one-dimensional distributions

t

(2.2.1) V t (r) = fni x (r)dx , r C Rt X Rt

Proof: Fix s E T. Consider a (0,1)-process y with initial

distribution 1s. Let V = Y By the constraction of (vtQ) the

u

process vt, t s has the same finite-dimensional distributions as

Vt, t t s. Both processes are right-continuous, therefore their ranges

have equal distributions. But the range of V. is equal to that of y.,

--8-

and that proves that M is (0,1)-generated (right-regenerativity is a

consequence of this fact).

By thu construction, the process vt - t is Markov with stationary

transition function and stationary one-dimensional distributions (equal

to V0 ). Hence M is a t.i. set. Any (0,1)-generated set is thin;

as a result, for the t.i. set M we have

(2.2.2) Q{t E MI = 0

Since M is (0,11)-generated

Q{z t E rIZ} = +{Y e r Is

+a.e. Q on the set {z t. Since A1 is bounded there exists s such

that A > s. The distribution of Z+ is equal to that of vs, and we

can write

(2.2.3) Q{zt E } = f P (dx)Q{zt E rlz+ = x)

>4f V (dx)Q{iY E r)

The last equality in (2.2.3) due to the fact that {z < tI = {z > 1.t

By virtu-e of Lemma 2.1 of [TJ the right hand side of (2.2.3) is equal

to

-9-

s t(2.2.4) fdyfnfy;, dx}j, Xx(dz)n(z; A2)

-= s Al

Let y= -Yt and let Q , x, n'(x; r), v*, etc. be defined as in Section 6

of [7]. Performing the same transformations as in Lemma 6.6 of 17], we

get that (2.2.4) equals

z

(2.2.5) f dxf n*(x;dz)fX(dy)n*(y; Rs )A2 AI s

By virtue of Lemma 2.1 of [7]

zf X* (dy)II*(; R )Q *f Y* < s}-s s

here a* is the first hitting time of ]-,s]. Hence (2.2.5) is equals

to (we use (6.11) of [7])

t(2.2.6) f dx11 ( A 2x; A2 f _ x(A1 x A2 )dx

Corollary: The distribution of -M is equal to that of M.

Proof: Since M is a (0,H)-generated set, z t is a Markov

process with the transition function p given by (5.2) of 17]. As a

result, vt is an entrance law with respect to p. Let v* be definedt t

as in Lemma 6.5 of [7]. Formula (2.2.6) shows that v* = vt. Repeating.4 t

the proof of Lemma 6.6 in our case, we get that z has backward transi-

tion function p*. Then we must argue in the same way as in Lemma 6.7

of [7].

-10-

Lemma 2.2.2. The set M satisfies (2.1.2).

Proof: In (2.1.2) we may consider only the functions f such that

(2.2.7) f(x,y) = 0 if x > y

Put R {(x,y): x t}, f = fl For A = rl,r2 rst st R 1, 2, rk

stset RA = Rrlrl U Rr2r2 U ... U Rrkrk , fA = flRA. If rl,r2,...rk...

is a sequence of all rational numbers and A(n) = {rI ... r n , then

fA(n) + f for any function f subject to (2.2.7). Trivial computations

show that the function fA(n) is a linear combination of the functions

f st s < t. Since both sides of (2.1.2) are stable under linear opera-

tions and monotone passage to the limit, we have to verify (2.1.2) only

for the functions fst s < t.

Qf fst ( Y ' ) 1 = Q{l Yt}Y Y

= Q{f (zt)}

= vt (fSt)

t= fn x(f st)dx

= f dx

= Idxffst (x~x + y)fl(dy)-m 0

3. Stationary Markov Processes with Infinite Underlying Distributionsand their Subprocesses.

3.1. Consider a (generalized) stationary Markov process (xt,P),

that is a process satisfying the definition given in Section 1.2 of [8].

Assume that xt is conservative, i.e. P{a # -1} = P{B # +-) = 0. Suppose

that the state space E ofthis process is divided into two sets D and

V in such a way that

(3.1.1) M = {t: xt E V} is closed a.e.

We denote by ]y,6[ an element of the set of all open intervals contiguous

to M. For each path x. and each ]Y,6[ we associate a trajectory w6

in D by the formula wy(s) = x , Y < s < 6. The set of all trajectories6e

r

in D with random birth time a and death time 0 is denoted by W. If

M satisfies 1.2.ci of [8] then it is possible to define a measure P on W

in the following way (W is endowed with the Kolmogorov a-field G).

(3.1.2) P{A) = }fil (w N

The process (w(s),P) is called a subprocess in D of the process (xt,P).

Let v, p, P be respectively the one-dimensional distribution, the transi-x

tion function and the transition probabilities of the process (xt,). The

formula (1.2.2) of [8] shows that P is a Markov measure with the transition

function p defined by l.l.y. If the measure v is a-finite, then so is

P, and if for each t

(3.1.3) {xt E V} =0

- V..4* I.

-12-

then the one-dimensional distribution of P is equal to that of P (namely

to v). In the sequel we shall consider only processes (xt,P) subject to

(3.1.3). Put s = inf {t>s: x t EV}; T =T 0 If

3.1.A. For each x E D

Px fT > t) 0 as t

then for each x E D

(3.1.4) p(tx; D) 0 as t -

If

3.1.B. For any set r C D such that v(r) < m

Rx s E r, T > s} 0 as s co

then for any set r such that P{w(0) e r) <

(3.1.5) F{w(s) E r, a < 0, 8 > s) 0 as s

If

3.1.C. For some s • 0

Ph < s) <

;.1 then

(3.1.6) P{Q 0, 0 < 8 s) <

:I

-13-

Let Tt be the semigroup generated by the transition function p.

Note that (3.1.4) is true iff Tt is a SD-semigroup. The condition (3.1.5)

holds iff v is null-excessive measure; (3.1.6) is true iff v is quasi-

finite excessive with respect to Tt measure. If both (3.1.5) and (3.1.6)

are satisfied then we say that the process (w(s),P) has a null-quasi-

finite one-dimensional distribution.

Let (0 be the sample space of the process (xt,P) and F be the

basic a-field in n on which the measure P is defined, and which

is supposed to contain all sets of P-measure zero. Denote by F the5

completion with respect to P of o(xu,u < s) and by Cs the completion

with respect to P of the a-field generated by the sets

{'u < r) , u,r < s

(If the process xt is regular, then Cs CF 3 .) We say that the set D

is regular for (xt,F) if for t > s, Cs V F and xt are conditionally

independent given x . (This definition certainly assumes Cs C F).

A Markov process (x1 ,) with the state space E = D U V and ati 1 1

Markov process (xt,Q2 ) with the state space E = D 2

equivalent, if the one-dimensional distributions of both processes are con-

centrated on D and their finite-dimensional distributions coincide.

The following theorems are similar to Theorems 1 and 2 in [8].

Theorem 3.1.1. Let (w(s),P) be a stationary Markov process in

the state space D with the transition function p, subject to (3.1.4).

If the one-dimensional distribution of P is null-quasi-finite, then this

process is a subprocess of a concervative stationary Markov process (x tP)

satisfying 3.l.A - 3.1.C for which D is a regular set.

Just as in [8] the set of all stationary Markov measures with

transition function p is denoted by S(p).

Theorem 3.1.2. If (w(s),P) satisfies the conditions of Theorem

3.1.1 and if in addition P is a minimal element of S(p), then the

process (xt,P) is unique up to equivalence.

3.2. In this section we prove Theorem 3.1.1. Consider the one-

dimensional distribution v of (w(s),P). It was proved in [31 that

(3.2.1) V = fVSds0

where v is an entrance law for p. We denote by P* a Markov measure

on G with the transition function p and the one-dimensional distribu-

stions v . Put

(3.2.2) n(r) = P'{ E .

Suppose that the process (xt,P) is constructed and M is defined by

(3.1.1). The same heuristic arguments as in Section 3.1 of [8] show that

the set M must be translation invariant (0,1)-generated and all the

cuts w6 must be conditionally independent, when M is fixed.

The next three lemmas prove that n, defined by (3.2.2), satisfies

-- (2.1.1).

Lemma 3.2.1. For any u > 0 the measure v - vT is finite.u

Proof: By our assumptions M S v - vT is a finite measure fors

some s >0. For each rO > jT r

-15-

(3.2.3) v - vTks = - vT + vT -T + "'' + VTck 1 ) - vTk

+ UTs + ... + VT(k-l)s

Each summond in the right side of (3.2.3) is a finite measure; and

so is v - vTks.

By virtue of (3.2.1)

ut

(3.2.4) v - vT = fv dt0

Hence if u < ks, then v - vT u < - T ks and the lemma is proved.

Lemma 3.2.1 shows that vS(D) is finite for m-almost all s > 0

(m is the Lebesque measure). On the other hand for t > s

vt(D) = v t(1) = vSTtsl vs(l) = vS(D)

Therefore vS(D) is finite for all s > 0 and is a decreasing function

of s. Consequently

(3.2.5) P*{B > s} = P*{w(s) E D) = VS(D) < , s > 0

Formula (3.2.5) shows that the restriction on any interval ]s,.] of

the measure R, defined by (3.2.2), is a finite measure; as a result,

n is a-finite.

Lemma 3.2.2. The measure R defined by (3.2.2) satisfies (2.1.1).

Proof: Put f(s) = vs(D).

-16-

(3.2.6) 7x A 1l(dx) - fxn(dx) + f(l)o 0

ffdx(dy) + f(l)C

where C = {(xy): x > 0, y > 0, x + y < i}. By Fubini's Theorem (3.2.6)

equals

11 1

f{fn(dy)ldx + f(1) = f(n(Rx) _ A(R 1 ))dX + f(l)0y 0

1

= f(f(x) - f(1))dx + f(1)0

1 1= ff(x)dx = fvX(D)dx

o 0

By virtue of (3.2.4) the right side of the above formula is equal to

(v - vTI)(D). Lemma 3.2.1 implies that this expression is finite.

By virtue of Theorem 2.1.1 and Lemma 3.2.2 we are able to construct

-' a (O,H)-generated translation invariant set M, subject to (2.1.2).

Let 0 be the corresponding sample space and Q be the corresponding

measure.

Lemma 3.2.3. For any function f on T X T

(3.2.7) P{f(a,8)1 =Q{lf(y,6)

Proof: Let Pt be defined by (3.2.1) of [8]. By formula (3.2.2)

* t of [8]

(3.2.8) P{f(=,B)) =JPIf(a,0)}dt

fP {f(t,B)1dt

- JP*{f(t,B + t))dt

- ({Jf(t,y + t)n(dy)ldt_..o 0

By virtue of (2.1.2), the right side of (3.2.8) is equal to the right

side of (3.2.7).

Consider a measure N on T x T x W defined below.

N(r x A x A) = P{a E r,O E A,w EA , r,A C T,A E G

Put

(3.2.9) N(B) = i(B x W) , B CT x T

Lemma 3.2.4. The measure N, defined by (3.2.9) is a-finite.

Proof: If n satisfies (2.1.1), then for t > 0

17 (Rt) <

The support of measure N is the set C = {(x,y): y > x). The set C

may be represented as a countable union of rectangles R ]u,v[ x ]r,q[,

where u < v < r < q. For such rectangle R

-18-

N(R) = Pfu < a < v, r < 8< q)

v= fP:{r < 0 < qidt

Uu

v

< fPf{8 > rldtu

v

< fP > r - vdtu

= (v - u)H(R r- v )

< @

Further steps in the construction of (xt,P) do not differ from the

analogous ones in [8]. We take the stochastic N-quasi kernel m(x,y; A)

which is a Radon-Nikodym derivative of N(dx x dy x A) with respect to

N(dx x dy). Then we define a sequence of stochastic Q-quasi kernels

nk(w; A) in the same way as it was done in Lemma 3.3.2 of [8]. We put

= 9 x W and define P on &I by the formula (3.3.3) of [8] (it is

necessary only to replace P in the right side of (3.3.3) by Q). To

justify the existence of such a measure P, we use Theorem 3.3.1

of [8], which is true for a-finite measure Q as well. We take

E = D U V, where V is a singleton, and put xt(w) = xt(wwlW2,..) V

* I if t w M(w) and we put xt(w) = wk(t)(t) otherwise (see the end of

Section 3 of [8] for details).

Lemma 3.2.5. The process (x ,P) is a conservative stationaryt

Markov process. The subprocess in D of (xt,,) is equal to (w~s),P).

-19-

To prove Lemma 3.2.5 we have to repeat without variations all the

arguments of Section 4 of [8].

Lemma 3.2.6. The set D is a regular set for (xt,f).

Proof: Let ulU 2,.. ..,uk, v11 v2 '.. .VTk, sls 2 ,... ns < s < t. We

need to show that for each r,rl,...,r t. .' there exists a function g

on E such that

(3.2.10) Plx s e r, x E r1,. . n n xt E A, T < Vl,.".,T < Vns s1 n ul un n

ZZ P{g(x ); x E r', 6 c- r 1,...,x 5 C6 r' ,T < v .. T

-20-

(3.2.11) {x sI E rl,x s E r,xt e A'Tu < v}

= 1 dl(U,V)) 2(s,t)1ABC(w2)

+ 61(u,v) A2(sl,sl )iA(w2 )XA3(s,t)iBC(w3)

+ 1 ( U,V)X 2 (sl,s)1AB(W2 ) X3 (t,t)1c(W 3)

Y6 1(u,v)x 2 (sitsI)I A (W2 )X 3 ( sl iB(w 3 ) A(t t)ic(W4)¥1

-21-

(3.2.13) P{ (a)*(ws)); AIB,a < s

-22-

Applying successively the Markov property of (w(s),P), Lema 4.1.1 of [8]

and Lema 6.8 in [T], we get

(3.2-1T) f*(x)v(dx) - Pfil (w(B))*(W(S)))r

= P~l r(w(p)),e(B))

= Q{Xm(y,a; u(s) e rme6)

Y

where

e'(y) = m(yryr- ;w(P) E. r)1rGj

In view of Levma 4.. in [8], (3.2.17) equals

(3.2.18) P{e'(a)l A(w(u))) - P~e'(a)&(Q); 0 u)

where

i() Pfwv(u) E- A)1P{BO >Y

-23-

where

E I(y) =V C)Y

-214-

If P*{W} = , then we must consider the local time Et of (xt,P)

at V. We have to introduce a filtration At = Ct V Ft, with respect to

which the local time E is adapted. Repeating Lemmas 5.4.1-5.h.3 and

5.5.1-5.5.2 in our case, we obtain the expression (5.5.3), which is true

in the case of infinite underlying distribution P as well. (The proofs

of the above lemmas were based on the lemmas and theorems of Chapters h

and 5 in [7]. The whole theory in [7] was developed under the assumption

that the underlying measure is a probability one. Nevertheless everything

remains the same in the case of infinite underlying distribution.) Then

we have to consider the process ys, which is the right-continuous inverse

of the local time E.. Repeating the proofs of Lemmas 5.6.1-5.6.3, we

get that yt - ys and As are independent; therefore (ys,P) is the

process with independent increments, whose Levi's measure can be obtained

through P*. Lemmas 5.7.1 and 5.7.2 of [8) are also true in the case

of infinite underlying distribution, and they show that the two-dimensional

distributions of (xt,P) are uniquely determined by the measure P.

4. Enhancing of Semigroups

4.1. Now we consider the semigroup Tt and the measure v des-

cribed in Theorem 1.

If Tt is a S-semigroup then there exists a transition

function p(t,x; I such that

Ttf(x) = p(t,x;f)

(see [4], Theorem 2.1). If Tt is a dying semigroup then

p(t,x;D) 0 as t -.

-25-

By the theorem of Kuznecov (see [5]) there exists a stationary Markov

process (w(s),P) with random birth and death times whose one-dimensional

distribution is equal to v and transition function is equal to p. (To

construct such a process (w(s),P) we need also to specify an excessive

function h, but in our case h(x) = 1.) The conditions of Theorem 1

imply that the one-dimensional distribution of P is null-quasi-finite.

By Theorem 3.1.1 there exists a covering process (xt,P) with the state

space E = D UV.

Let p(t,x; r) be the transition function of P. The same way

as in 19] we can show that the condition 1.2.D implies

(4.1.1) p(t,x; D) = 1 for all t and x E D

Therefore p(t,x; V) E 0 and p(t,x; -), considered as a kernel from D

into D, is a transition function. By Theorem 3.1.1 (xt,P) is a stationary

conservative process with the one-dimensional distribution v; consequently,

v is invariant with respect to p. The semigroup Tt generated by

is the semigroup we are looking for. The properties 1.2.A-1.2.C are

automatically satisfied by any semigroup generated by a transition func-

tion. The property 1.2.E' is a consequence of (4.l.1); and (1.2.1) follows

from the fact that (xt,P) is a covering process for (w(s),P), for which

l.l.y holds. Lemma 3.2.7 gives us the explicit expressiong for T tf(x)

in terms of "internal" characteristics of v and T (we make trivialt

transformations in (3.2.16) to obtain the formula below).

(4.1.2) Ttf(x) = Ttf(x) + fQN{ft-YvYs(f)dslpx(dy)tt 0 0

-26-

where ix is a measure on ]0,m[ such that pir,u] = Trl(X) - ul(W ,

the family- v is an entrance law with respect to Tt for which (3.2.1)

holds, and (y sQ0) is an increasing process with independent increments

with translation constant 0 and the Levi measure R such that

nIr,u] = Vr(D) - vU(D)

It is interesting to compare the formula (4.1.2) with the result

of Getoor (see [4], Theorem (8.1)). He solves the inverse problem, namely,

he finds an invariant distribution v for the transition function

given by the formula analogous to (3.2.16). The expression for v he

obtains is similar to (3.2.1).

4.2. The proof of Theorem 2 does not differ from the proof of

Theorem 2 in [9]. We have to consider a conservative stationary Markov

process (xt,P) with the one-dimensional distribution v and the transi-

tion function p which generates the semigroup Tt (but in contrast

to the situation in [9], now P may be an infinite measure). A multi-

plicative functional at is constructed in such a way that

p(s,x; r) = P xfr(xs)ms}

Let n be the sample space of the process xt . We put n = 0 x (T)"

and construct a measure Q on a and a family of random variables Ts(z)

in such a way that

4.2.A. The marginal distribution of Q on 0 is equal to P.

-I ''., ,.. .. , ..

-27-

4.2.B. For t > s the conditional probability of T s to be greater

than t, given w is equal to ats (e w).

4.2.C. For t > s Ts- -t on the set {T s > t).

4.2.D. The 0-field Cs V F and the pair (xt,T t ) are conditionally

independent, given xs , where Cs is the minimal a-field in S1 generated

by the sets Tr < u; r,u < s

The family T has the same properties as the family of the first

hitting times of a set in the state space. We put M(w) to be the closure

of the set of values of the function T.(w). We put x*(w) to be equal to

xt(w) if t E M(W) and x*(w) E V otherwise. In the same way as in [9]

one can show that (x*,Q) and (x P) has the same finite-dimensional

distributions and the subprocess in D of (x*,Q) is equal to (w(s),P)

where P is a Markov measure with the one-dimensional distribution v and

the transition function p. Now we need only to apply Theorem 3.2.1 to

obtain the final result.

1 ' . '

-28-

References

I [11 Dellacherie, C. [1972], Capacitds et Processus Stochastigue, Berlin-Heidel-berg-New York: Springer

[2] Dynkin. E.B. [1965], Markov Processes, Vol. 1 & 2, Berlin-Heidelberg-NewYork: Springer.

[3] Dynkin, E.B. [1980], "Minimal Excessive Measures and Functions," TAM~258, no. 1, 217-2144.

[4i] Getoor, R.K. [1979], "Excursions of a Markov Process," Annals of Proba-bility, 7L, no. 2, 2144-266.

[5] Kuznecov, S.E. 11973], "Construction of Markov Processes with Random Timesof Birth and Death," Theory of Probability and its Applications,XVIII, 571-575.

[6] Maisonneuve, B. [1981], "Ensembles Regeneratifs de la Droite," (preprint,forthcoming in Z. fur W. Theorie und Ver. Geb.).

[7] Taksar, M.I. [1980], "Regenerative Sets on Real Line," Seminaire deProbabiliti~s XIV. Lecture Notes in Mathematics,volume 7814,Springer.

* [8] Taksar, m.i. [1981], "Subprocesses of Stationary Markov Processes," Z. fur-W. Theorie und Ver. Geb., 55, 275-299.

[9] Taksar, M.I. [1980], "Enhancing of Semigroups," (preprint).

REPORTS IN THIS SERIES

1wi ''lhe St I u, lire aild Slahili1t N it rI,-tpetiflive l)vrtastncal Sysierns,' by D~avid (ass arid Karl Shell.lt'l %A~lol~ulsti.( I. inetilimli arid file J.airIal Maliker.' hi, J. I .Slighjtl.Lo., , I Le (is lpkiratIlot I 1s .1- sI J I Stsg t/~It, 'Alctiimt ri Resinm i i Stale IIIt ite \ggtegitt. aiid the Scale I:tlIII Public (ootds," by D~avid A. Starrett.

lr'4 '~I ,IAs )It ild Regulaition - bs %lik:hel Sence.I ~ ' "I,~\.s lijgel . mris[ I'ii it]s, i -del kit Borrowinig.'' bys Duncain K. Ir oley arid Maim) F. liellwig.

Iso Iit, l i I eI Ii Is. R isk It,! I it or Iai i \ - I %,-1ek I ow a Ids a I lieo ry a I ItierIa ity -by Jose ph L. St ight /.14 ' s ii 1~ \jali~is It fihe DI st iioits )I I stuiies fit Simultaneous Equations far Alternative Parameter

It.s *'I siIniji'-1 1 Illj i all-t, ii, Relinkinships .. ppromtsie D~istributioins and (onlections with Simultaneous

It,"t 'AI,,iwjsit ii! IIC, Rate i.i I kii,., lii ot Lxliusttble Resources.'" h\' Joseph E. Stighti.I 'It ' quilhibI~iiiIII it .'tstt'estiIe lttnturite Maikets Ait Ussay on the Ecoinomtics of Intpertect Inforniation.'' by

II "St tong ttsei- i I east Sqfuares I timates itt Norisal Linear Regression," by T. W. Anderson and John B. Taylor.11 1'elncitse S tierieN ituidet ll efettil h1ltoittatiost Structures. Ar: Application to I ade Policy," by Partha Dasgupta

alid ii w',l sttlit/I I[lie Incidemie and, I tticicli,\ I ite~ ts of Trt\c, itt Income frotit (apitaL.' by John B. Shoven.1 "4. ''listriuton ,If a "lamituti Likelti id Ilstimtate o~f a Slope (tteTici_ nt: Tlte [.IML Estimate for Known Coivariance

Matst\.. his I A. *lidclsoil And lila.,ilii S.issl75 "N L iitIII otile lost ot Oscridenit ti ts Rest rictions." by Josephs B. Kadane and T. W. Anderson.I 7t). ''Ai Asy I piotic I x pattsionfl ilie i stribio h ot t ile Maximttuma Likelihiootd Estintiite of rthe Slope Coefficient fin a

177. 'Sottte I- .'peuimisiil Results tat the Staiistical Prttperties at Least Squares Estimates in Controil Problems." bysI. W.. Atides sotn arid Johnir B. laylIms

I178. "A Note til "Eulfilled Lxpectatitits" Lquililit." by David M. Kreps.179. "Uncertaint\ anid the Rate ot Extractioit under Alternative Institutional Arrangemtents," by Partha Dasgupta andI~ Josephi E. Stiglit,.180. "Budget Displacemtent Effects oi lttionary Finance." by Jerry Green and E. Slteshinski.

181. "Towards a Marxist Theory of Money," by Duncan K. Foley.182. "The Existence of Futures Markets, Noisy Rational Expectations and Informational Externalities," by Sanford Grossman.183. "On the Efficiency of Cosipetitive Stock Markets where Traders have Diverse Information." by S:ifiord Grossman.184. "A Bidding Model of Perfect Comrpetition." by Robert Wilson.185. "A Bayesiait Approach it)r tlte Production of Information and Learning by Doing," by Sanford J. Grossman, Richard E.

Kihlstrotri and Leonard J. Mirnoan.186. "Disequilibrius Allocations and Rectintracting." by Jean-Miclsel Grandtniott Guy Laroque and Yves Younes.

187. "Agreeing to Disagree." by Robert J. Ausstann.188. "The Maximumr Likelihood and tlte Noinlinear Three Stage Least Squares Estimator in the General Nonlinear Simtultaneous

Equation Model," by Takeshi Atneiniya.189. "The Modified Second Round Estimator in the Genteral Qualitative Response Model." by Takeshi Amnemiya.190. "Some Theorems in tie Linear Probability Nlodel." by Takeshri Astezttiya.191. "The Bilinear Complementarity Problem and Competitive Equilibria of Linear Economic Models," by Robert Wilson.192. "Noncooperative Equilibrium Concepts for Oligopoly Theory," by L. A. Gerard-Varet.193. "Inflation and Costs of Price Adjustmient." by Eytan Shieshinski and Yoramt Weiss.194. "Power and Taxes in a Multi-Consinitdity Economy," by R. J. Autnann and M. Kurz.195. "Distortion of Preferences. Incomte Distirburirin and the Case for a Linear Incomste Tax." by Mordecai Kurz.196. "Search Strategies for Nonrenewable Resource Deposits." by Richrard J. Gilbert.197. "Demand for Fixed Factors. Inflation and Adjustmenrt Costs," by Eytan Sheshinski and Yoratts Weiss.198. "Bargains and Ripoffs: A Model of Monoporlist ically Comtpetitive Price Dispersions," by Steve Salop and Joseph Stiglitz.199. "The Design of Tax Structure: Direct Versus Indirect Taxation by A. B. Atkinsnon and J. E. Stiglitz.200. "Market Allocations oif Location Chouice itt a Model with Free Mobility." by David Starrett.201. "Efficiency in the Optimum Supply (if Public Goods." by Lawrence J. Lau. EyItan Sheshinski and Joseph E. Stiglitz.

202. "Risk Sharing. Sharecripping and Uncertain Labor Markets." by David M. G. Newberry.a203. "On Non-Walrasian Equilibria," by Frank liahn.

204. "A Note on Elasticity (if Substitution Functions." by Lawrence J. Lau.205. "Quantity Costsraints as Substitutes for Inoperative Markets: Thre Case of tile Credit Markets." by Mordecai Kurz.

206. "Incremental Cornsumer's Surplus attd Iledonic Price AdjustmIlent,." by Robert D. Willig.

REPORTS IN THIS SERIES

207. 'Oplional Leclm tono it it iit Sttck, .'' b Ricia i d Gilbert.208. ''Sonme ~VlInII il (IIII 1-Sq Ua e I Si11aI .111 ad (Xili pa isoIls of Normal anid Logist icaModels in Qualitative Response

200. ''A ('harac etuiatt i it* tire Opt m al, ,I of E ilibrini in I ncomtplete Mar ket s.'' by Sanford J. Grossman.210. "Inflatioin arid faxes itI a Giwn , oiin %iiDb arid Equity Finance." by M. Feldstein, J. Green and E. Sheshinski.211, ''The Specification anid listiiatioii of a Mult vat ate Logit Model.'' by Takeshi Ainemiya.212. Prices anid Queues as Scicciting Devices III COiipetitivc Markets." by Jiiseph E. Sliglitz.213. "Conditions for Stronig ('otsiSIeticy of Least Square% Estiiiates in Linear Models," by T. W. Anderson and John B. Taylor.214. -Ut ilit arianisim arid HoIiriuzintal Eq ii\I t [ie (ase forI Randotm Taxat ioini,' by Joseph E3. St ig i tz.2 15. "'Simiiple Forimulae lbor Opt itial I nco tie Ta xat itn and [Ile .Measuremnent iif Inequality.'' by Joseph E. Stiglitz.216. "Teitporal Resolution of lUnei tait anid Dynamiic Chioice Behavior," by David M. Kreps anid Evan L. Porteus.217. "'Tire Estitiatioti(of Notilintear labor Suippl, 1-tiictll usWithi Taxes from a Truncated Sample.'' by Michael Hurd.218. ''Tire Weltii e Imiiplicatioiis oftlle UnteiitploN int Rate."' by Mlichtael Ifurd.21 1f. "Keytiesian Economiics arid Getneral Equilibrimn Theory: Reflections oin Some C'urrent Debates," by Frank Hahn.220. "Tire (ore oft an Exchange Economiy ws ith Differenttial Inli mation.'' by Robert Wilson.221. ''A Comnpetitive Mi del of Excltatge.'' by Robert Wilson.222. "iter mediate P'references anid thle Majority Rule." by Jeati-Michel Grandmiont.

2 23. "'Tire Fixed Prtce Equilibria. Somue Results III Local (ottparatise Statics.'' by Guy Laroque.224. "Onl Stockliolder litantittity ini Makinig ProuitiC1 arid Finatial D~ecisioins." by Sanf ird J. Grossman and Joseph E. Stiglitz.

"2. Selection of Regressoys." by Ilakeshi Anieiia.

''. A Note oni A Randtiii (oefticients Model. .'' bI lakcslii Aieniiya.2 '.'A Note onl a I leteroscedastc Model.' by, Litkeslt Aiiiciiiia.

228. ''Welfare Measrtemtent for local Publlic: titatice.'' h\ David Starrett.'2k. ''lteitiployvmntt Equtlibrium %kithi Rationial Expectations.'' by W. P. Ifller arid R, M. Starr.

230. ''A Thtemr of"(ottpetitive Eqilibriutim III Stock Market Ecotoinies.'' by Santford J. Grossiman atid Oliver D. Hart.231. '"An Application of'Stciii's Mlet IiSi t tlie p~rolm oiif Single Periiid ('ontrol of Regression Models.'' by Asad Zainan.

23. "Seodc BetWae Ecnms i ii~edLcoioodhsiin of Da i Slop Cefrcintt.b233 "ireLoicof tire Fix-Price Met hod.'' by Jeat-i-Nhiel Granditont.

234. "Sae o BetWae Ecrinontics n the Mltnie liotod E"stimavid Sftaeti.eCefiin n Aprxn os"b

T. W AndrsonandTakaiiitsu Sawa.

237. '"The Est iiiat ion oif a Si iilltanceitts- [qualtit n (;icrldPobit Model by Takesti

238. "The ('onsistency, of the Maximumii Likelihood Estimiator in a IDiscquilibriuit Mrodel." by T. Ametniya and G. Sen.239~. "Numerical Evaluation of the Exact arid Approximate D~istributiont Fumnctions of the Two-Stage Least Squares Estimate."

by T. W. Anderson and Takaititsu Sawa.240. "Risk Mleasuremtentt of Public Projects." by Robert Wilsotn.241. "On the Capitaiiation I lypotliesis in ('liised ('otmmunities." by David Starrett.242. "A Notte oni the Urtiqueness of the Represetntationt of Cotntiidity-Augtmenling Technical Change," by Lawrence J. Lau.243. "*Thre Property Rights Doctritne anid Demiartd Revelation tinder Incomplete Information," by Kenneth J. Arrow.244. "Optimal Capital Gainis Taxationi Unider Limited Infitriatioti." by Jerry R. Green and Eytan Sheshinski.245. ''St raighitforward Individual Inicentive Comnpatibility in Large Econonmies.'' by Peter J1. liatmimond.246. "On the Rate oif (Convergence oftlle Core." by Robert J. Altmatnn.247. "Unsamisfactir Equilibria." by Frank I lahit.248. "Existence ('onditions for Aggregate Demttand Functioins: The Case of a Single Index," by Lawrence J. Laut.249. "Existence C'onditiotns for Aggregate Dettatnd Functions: Tile ('as: of Multiple Inee, by Lawrence J. Laut.250. "A Note on Exact Index Numbers," bly Lawrence J. Latt.251. "Linear Regression Using Both Temporally Aggregated and Temporally Disaggregated Data," by Cheng lisiao.252. "*The Existence of Ecottotitic Equilibria: Conitinuity and Mixed Strategies." by Partha Dasgupta and Eric Maskin.

253. "A C'ttmplete Class Thteoremt for thle ('ontrol Probslem and Further Results on Admtissibility and Inadmissibility," by Asad Zaman.2154. "Measure'Ilased Values til Ma13rket ("amles." by Sergiu liart.255. "Altruismt as an Outcome of~ Social Interaction." by Mordecai Kurz.256. "A Representation Tisrei ftir 'Preference fur Flexibility'," by David M. Kreps.

257. "Time Existence tif' Efficient atnd Itncentive (ottpatible Equilibria Witli Public Gooids." by Theodore Groves andJoihtt 0. ILedyard.

258. "Efficient Ciillective Choice Wilit 'oinpensation." by Theodore Groves.

259. "On tire Imtpoissibility of inforitatitinatty Effienrt Markets." by Sanford J. Grossman and Joseph F. Stighitz.

: REPORTS IN THIS SERIES

260. "Values for Games Without Sidepayments: Some Difficulties With Current Concepts," by Alvin E. Roth.

261. "Martingles and tile Valuation of Redundant Assets," by J. Michael Harrison and David M. Kreps.

262. "Autoregressive Modelling and Money Income Causality Detection," by Cheng Hsiao.

263. "Measurement Error in a Dynamic Simultaneous Equations Model without Stationary Disturbances," by Cheng Hsiao.

264. "The Measurement ot Deadweight Loss Revisited," by W. E. Diewert.

265. "The Elasticity otf Derived Net Supply and a Generalized Le Chatelier Principle," by W. E. Diewert.

266. "Income Distribution and Distortion of Preferences: the 2 Commodity Case," by Mordecai Kurz.

267. "The -1"2 Order Mean Squared Errors of the Maximum Likelihood and the Minimum Logit Chi-Square Estimator," by Takeshi Amemiya.

268. "'Temporal Von Neumann-Morgenstern and Induced Preferences:' by David M. Kreps and Evan L. Porteus.

269. "Take-Over Bids and the Theory of the Corporation," by Stanford Grossman and Oliver D. Hart.

270. "The Numerical Values of Some Key Parameters in Econometric Models," by T. W. Anderson, Kimio Morimune and Takamitsu Sawa.

271. "Two Representations of Information Structures and their Comparisons," by Jerry Green and Nancy Stokey.

272. "Asymptotic Expansions of the Distributions of Estimators in a Linear Functional Relationship when the Sampie Size is Large," byNaoto Kunitomo.

273. "Public Goods and Power," by R. J. Aumann, M. Kurz and A. Neyman.

274. "An Axiomatic Approach to the Efficiency of Non-Cooperative Equilibrium in Economies with a Continuum of Traders," by A. Mas-Colell.

275. "Tables of t,' Exact Distribution Function of the Limited Information Maximum Likelihood Estimator when the Covariance Matrix inKnown," by T. W. Anderson and Takamitsu Sawa.

276. "Autoregressive Modeling of Canadian Money and Income Data," by Cheng Hsiao.

277. "We Can't Disagree Forever," by John D. Geanakoplos and Heraklis Polemarchakis.

278. "Constrained Excess Demand Functions," by Herklis M. Polemarchakis.

279. "On the Bayesian Selection of Nash Equilibrium," by Akira Tomioka.

280. "Disequilibrium Econometrics in Simultaneous Equations Systems," by C. Gourierous, J. J. Laffont and A. Monfort.

281. "Duality Approaches to Microeconomic Theory," by W. E. Diewert.

282. "A Time Series Arntlysis of the Impact of Canadian Wage and Price Controls," by Cheng Hsiao and Oluwatayo Fakiyesi.

283. "A Strategic Theory of Inflation," by Mordecai Kurz.

284. "A Characterization of Vector Measure Games in pNA," by Yair Tauman.

285. "On the Method of Taxation and the Provision of Local Public Goods," by David A. Starrett.

286. "An Optimization Problem Arising in Economics: Approximate Solutions, Linearity, and a Law of Large Numbers." by Sergiu Hart.

287. "Asymptotic Expansions of the Distributions of the Estimates of Coefficients in a Simultaneous Equation System," by YasunoriI:ujikoshi, Kimio Morimune, Naoto Kunitomo and Masanobu Taniguchi.

288. "Optimal & Voluntary Income Distribution," by K. J. Arrow.

289. "Asymptotic Values of Mixed Games," by Abraham Neyman.

290. "Time Series Modelling and Causal Ordering of Canadian Money. Income and Interest Rate," by Cheng Hsiao.

291. "An Analysis of Power in Exchange Economies," by Martin J. Osborne.

292. "Estimation of the Reciprocal of a Normal Mean," by Asad Zaman.

293. "Improving the Maximum Likelihood Estimate in Linear Functional Relationships for Alternative Parameter Sequences." b ) Kini'oMorimune and Naoto Kunitomo.

294. "Calculation of Bivariate Normal Integrals by tile Use of Incomplete Negative-Order Moments," by Kei 1 akeuchi and Akmchi I akemura.

295. "On Partitioning of a Sample with Binary-Type Questions in Lieu of ('ollecting Observations." by Kenneth J. Arrow.. Leon Pesotchtnskvand Milton Sobel.

296.

297. "Tie Two Stage Least Absolute Deviations Estimators." by Takeshi Amernii.

298. "Three Essays on Capital Markets," by David M. Kreps.

299. "Infinite Horizon Programs," by Michael J. P. Magill.300. "Electoral Outcomes and Social Log-Likelihood Maxima." by Peter (oughlin and Shmuel Nitzan.

301. "Notes on Social Choice and Voting." by Peter Coughlin.

302. "Overlapping Expectations and Hart's Conditions for I-qutlibrium in a Securities Model," by Peter J. Hammond.

301 . "Directional and Local Electorate Competitions with Probabilistic Voting." by Peter (oughlin and Shmuel Nitzan.

304. "Asymptotic Expansions of the Distributions of the Test Statistics fot Overidentifying Restrictions in a System of SimultaneousEquations," by Kunitomo, Morimune, and Tsukuda.

305. "Incomplete Markets and the Observability ot Risk Preference Properties," by H. Ii. Polemarchakis and L. Selden.

306. "Multiperiod Securities and the Efficient Allocation o Risk A (omment on the Black-Scholes Option Pricing Model." byDavid M. Kreps.

307. "Asymptotic :xpansions of the Distributions of k-('lass I- stimators Ahen the Disturbances are Small." by Naoto Kunitomo,Kimio Morimune. and Yoshihiko Tsukuda.

308. "Arbitrage and Equilibrium in Economies with Ininimtel Man) (ommodities," b David M. Kreps.

309. "Unemployment Equilibrium in an Economy with linked Prices." by Mordecai Kur,.

310. "Pareto Optimal Nash Equilibria are ('otnopetitisv in a Repeated I- k.onom%.," b Motrdecai Kuri and Sergmu Hart.

311. "Identification." by Cheng Hsiao.

312. "An Introduction to Yso-Person Zero Sum Repeated Games with Incomplete Information." bs Sylain Sormn

Reports in this Series

313. "Estimation of Dynamic Models With Error Components," by T. W.Anderson and Cheng Hsiao.

314. "On Robust Estimation in Certainty Equivalence Control," byAnders H. Westlund and Hans Stenlund.

315. "On Industry Equilibrium Under Uncertainty," by J. Dr ze andE. Sheshinski.

316. "Cost Benefit Analysis and Project Evaluation From the Viewpointof Productive Efficiency" by W. E. Diewert.

317. "On the Chain-Store Paradox and Predation: Reputation forToughness," by David M. Kreps and Robert Wilson.

318. "On the Number of Commnodities Required to Represent a MarketGames," Sergiu Hart.

319. "Evaluation of the Distribution Function of the LimitedInformation Maximum Likelihood Estimator," by T. W. Anderson,Naoto Kunitomo, and Takamitsu Sawa.

320. "A Comparison of the Logit Model and Normal DiscriminantAnalysis When the Independent Variables Are Binary," by TakeshiAmemiya and James L. Powell.

321. "Efficiency of Resource Allocation by Uninformed Demand," byTheodore Groves and Sergiu Hart.

322. "A Comparison of the Box-Cox Maximum Likelihood Estimator andthe Nonlinear Two Stage Least Squares Estimator," by TakeshiAmemiya and James L. Powell.

323. "Comparison of the Densities of the TSLS and LIMLK Estimatorsfor Simultaneous Equations," by T. W. Anderson, Naoto Kunitomo,and Takamitsu Sawa.

324. "Admissibility of the Bayes Procedure Corresponding to the UniformPrior Distribution for the Control Problem in Four Dimensions butNot in Five," by Charles Stein and Asad Zaman.

325. "Some Recent Developments on the Distributions of Single-EquationEstimators," by T. W. Anderson.

326. "On Inflation", by Frank Hahn

327. Two Papers on Majority Rule: "Continuity Properties of MajorityRule with Intermediate Preferences," by Peter Coughlia and Kuan-PinLin; and, "Electoral Outcomes with Probabilistic Voting and NashSocial Welfare Maxima," by Peter Coughlin and Shmuel Nitzan.

328. "On the Endogenous Formation of Coalitions," by Sergiu Hart andMordecai Kurz.

329. "Controliability, Pecuniary Externalities and Optimal Taxation,"by David Starrett.

330. "Nonlinear Regression Models," by Takeshi Amemiya.

Reports in this Series

331. "Paradoxical Results From Inada's Conditions for Majority Rule,"

by Herve Raynaud.

332. "On Welfare Economics with Incomplete Information and the SocialValue of Public Information," by Peter J. Hammond.

333. "Equilibrium Policy Proposals With Abstentions," by Peter J.Coughlin.

334. "Infinite Excessive and Invariant Measures," by Michael I. Taksar.