foss lecture6

8/12/2019 Foss Lecture6

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Lectures on Stochastic Stability

Sergey FOSS

Heriot-Watt University

Lecture 6

Coupling again: the Renovation Theory

1 Introduction

A large fraction of the applied probability literature is devoted to the study of existence and unique-

ness of a stationary solution to a stochastic dynamical system, as well as convergence toward such

a stationary solution. Examples abound in several application areas, such queueing theory, stochas-

tic control, simulation algorithms, etc. Often, the model studied possesses a Markovian property,

in which case several classical tools are available. In the absence of Markovian property, one has

few tools to rely on, in general. The concept of renovating event was introduced by Borovkov

(1978) in an attempt to produce general conditions for a strong type of convergence of a stochastic

process, satisfying a stochastic recursion, to a stationary process. Other general conditions for ex-

istence/uniqueness questions can be found, e.g., in Anantharam and Konstantopoulos (1997,1999)

and in Diaconis and Freedman (1999). The so-called renovation theory or method of renovatingeventshas a flavor different from the aforementioned papers, in that it leaves quite a bit of freedom

in the choice of a renovating event, which is what makes it often hard to apply: some ingenuity is

required in constructing renovating events. Nevertheless, renovation theory has found several ap-

plications, especially in queueing-type problems [see, e.g., Brandt et al. (1990) and Baccelli and

Bremaud (1994) for a variety of models and techniques in this area].

Renovation theory is stated for a discrete-time stochastic recursive processes, i.e., random

sequences {Xn} defined by recursive relations of the form

Xn+1= f(Xn, n+1),

wheref is appropriately measurable function, and {n} a stationary random sequence. A standardexample of such a recursion is a sequence of Keifer-Wolfowitz vectors in 2-server first-come-first-

served queue which is defined by W0= (0, 0)and, forn= 0, 1, . . .,

Wn+1= R(Wn+ e1n itn)+

where e1 = (1, 0), i = (1, 1), R(x1, x2) = (x(1), x(2)) is the non-decreasing ordering, and(x1, x2)

+ = (max(0, x1), max(0, x2)). Here {tn} are inter-arrival and {n} are service times.

We take a fresh look at the renovation theory and formulate it for processes that do not necessar-

ily obey stochastic recursions. We give a self-consistent overview of (extended) renovation theory,

and strong coupling notions; second, we shed some light into the so-called coupling from the past

property, which has drawn quite a bit of attention recently, especially in connection to the Propp-

Wilson algorithm [see Propp and Wilson (1996)] for perfect simulation (alternative terminology:


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exact sampling). We do this, by defining strong forward and backward coupling times. We also

pay particular attention to a special type of backward coupling times, those that we call verifiable

times: they form precisely the class of those times that can be simulated. Verifiable backward times

exist, e.g., for irreducible Markov chains with finite state space (and hence exact sampling from the

stationary distribution is possible), but they also exist in other models, such as the ones presented in

the second part of the paper.

2 Strong coupling notions

We start with revisiting the notions of coupling (and coupling convergence) that we need. For

general notions of coupling we refer to the monographs of Lindvall (1992) and Thorisson (2000).

For the strong coupling notions of this paper, we refer to Borovkov and Foss (1992) and to Borovkov

(1998).

Consider a sequence of random variables {Xn, n Z} defined on a probability space (,F, P)

and taking values in another measurable space (X,BX). We study various ways according towhichX couples with another stationary process{ Xn, n Z}. We push the stationarity structureinto the probability space itself, by assuming the existence of a flow (i.e., a measurable bijection)

: that leaves the probability measure P invariant, i.e., P(kA) = P(A), for allA F,and allk Z. In this setup, a stationary process{ Xn}is, by definition, a-compatible process inthe sense that Xn+1 = Xn for alln Z. Likewise, a sequence of events{An}is stationary ifftheir indicator functions{1An} is stationary. Note that, in this case, 1An = 11An = 1An+1for all n Z. In order to avoid technicalities, we assume that the -algebra BX is countablygenerated. The same assumption, without special notice, will be made for all-algebras below.

We next present three notions of coupling: simple coupling, strong (forward) coupling and

backward coupling. To each of these three notions there corresponds a type of convergence. These

are called c-convergence, sc-convergence, and bc-convergence, respectively. The definitions below

are somewhat formal by choice: there is often a danger of confusion between these notions. To guide

the reader, we first present an informal discussion. Simple coupling between two processes (one of

which is usually stationary) refers to the fact that the two processes are a.s. identical, eventually. To

define strong (forward) coupling, consider the family of processes that are derived fromXstartedfrom all possible initial states at time 0. To explain what the phrase in quotes means in a non-Markovian setup, place the origin of time at the negative index m, and run the process forward tilla random state at time0is reached: this is the process Xm formally defined in (2). Strong couplingrequires the existence of a finite random time 0 such that all these processes are identical after. Backward coupling isin a sensethe dual of strong coupling: instead of fixing the starting time(time0) and waiting till the random time , we play a similar game with a random starting time

(time 0) and wait till coupling takes place at a fixed time (time 0). That is, backward couplingtakes place if there is a finite random time 0 such that all the processes started at times priorto are coupled forever after time 0. The main theorem of this section (Theorem 1) says thatstrong (forward) coupling and backward coupling are equivalent, whereas an example (Example 1)

shows that they are both strictly stronger than simple coupling.

We first consider simple coupling. Note that our definitions are more general than usual because

we do not necessarily assume that the processes are solutions of stochastic recursions.

Definition 1(simple coupling).

1) The minimal coupling time betweenXandXis defined by

= inf{n 0 : k n Xk = Xk}.


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2) More generally, a random variable is said to be a coupling time between XandX iff1

Xn = Xn, a.s. on {n }.

3) We say thatX coupling-converges (or c-converges) to X iff < , a.s., or, equivalently, iff


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Definition 2(strong coupling).

1) The minimal strong coupling time betweenXandXis defined by

= supm0

(m), where

(m) = inf{n 0 : k n Xmk = Xk}.

2) More generally, a random variable is said to be a strong coupling time (or sc-time) betweenXandX iff

Xn = X0n = X

1n =X

2n = , a.s. on {n

}.

3) We say that {Xn} strong-coupling-converges (or sc-converges) to {Xn} iff j) =, we have

that the latter infinite product is zero, i.e., = +, a.s. So, even thoughXcouples with X, it doesnot couple strongly.

Proposition 2(sc-convergence criterion). Xsc-converges to X iff

P

liminfn

m0

{Xn = Xmn }

= 1.

Proof. It follows from the definition of that

{


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Definition 3(backward coupling).

1) The minimal backward coupling time for the random sequence {Xn, n Z} is defined by

= supm0

(m), where

(m) = inf{n 0 : k 0 Xnm =X(n+k)m }.

2) More generally, we say that is a backward coupling time (or bc-time) forX iff

m 0 Xtm =X(t+1)m =X

(t+2)m = , a.s. on {t

}.

3) We say that {Xn} backward-coupling converges (or bc-converges) iff


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On the other hand, from the definition of (that is, the strong coupling time between Xand X), wehave:

{ n}=k0

{Xn+k =X0n+k =X

1n+k =X

2n+k = }.

Applying a shifting operation on both sides,

{n n}=k0

{Xk =Xnk =X

n1k =X

n2k = }. (6)

The events on the right hand sides of (5) and (6) are identical. Hence (4) holds for all n, and thusP(n) = P(n n) = P( n), for all n. Finally, ifP(


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allj 0. In this case, the-field G,m contains all eventsA such thatA {j } Gj,m, forallj 0.

Definition 4(verifiable time). An a.s. finite nonnegative random timeis said to be verifiable withrespect to an increasing family of-fields {Gj,m, j 0 m}, if there exists a sequence of

random times{(m), m 0}, with(m) being a backwards G,mstopping time for allm, suchthat:

(i) = supm0 (m),

(ii) For allm 0,Xnm =X(n+i)m for alli 0, a.s. on {n (m)},

(iii) For allm 0, the random variableX(m)m is G(m),mmeasurable.

Some comments: First, observe that ifis any backwards coupling time, then it is alwayspossible to find(m)such that (i) and (ii) above hold. The additional thing here is that the (m)are backwards stopping times with respect to some -fields, and condition (iii). Second, observe thatany verifiable time is a backwards coupling time. This follows directly from (i), (ii) and Definition

3. Third, define

m= max((0), . . . , (m))

and observe that

(Xt0 , . . . , X tm) = (X

t10 , . . . , X

t1m ) = , a.s. on {t m}.

Thus, a.s. on {t m}, the sequence (Xt0 , . . . , X

tm) does not change with t. Since it also

converges, in total variation, to( X0, . . . , Xm), where Xis the stationary version ofX, it followsthat

(Xt0 , . . . , X tm) = (X0, . . . , Xm), a.s. on {t m}.

Therefore,

(Xm0 , . . . , X mm ) = (X0, . . . , Xm), a.s.

Sincem (i), for each0 i m, we haveXmi =X

(i)i , and this is G(i),imeasurable

and so, a fortiori, Gm,mmeasurable (the -fields are increasing). Thus,(X0, . . . , Xm) is Gm,mmeasurable. In other words, any finite-dimensional projection ( X0, . . . , Xm)of the stationary dis-tribution can be perfectly sampled. That is, in practice,{Gj,m}contains our basic data (e.g., itmeasures the random numbers we are using),m is a stopping time, and(X0, . . . , Xm)is measur-able with respect to a stopped -field. This is what perfect sampling means, in an abstract setup,withoutreference to any Markovian structure.

Naturally, we would like to have a condition for verifiability. Here we present a sufficient

condition for the case where renovating events of special structure exist. To prepare for the theorem

below, consider a stochastic process {Xn, n Z} on (,F, P , ), the notation being that of Section2. Let{n = 0

n, n Z} be a family of i.i.d. random variables. For fixed Z, consider theincreasing family of-fields

Gj,m := (j, . . . , m).

Consider also a family {Bn, n Z} of Borel sets and introduce the events

Aj,m := {j B, . . . , m Bm+j}

A0:=m0

A0,m= { B, . . . , 0 B0, . . .}

An := {n B, . . . , nB0, . . .}= n

A0.


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Theorem 2(verifiability criterion). With the notation just introduced, suppose P(A0)> 0. SupposetheAn are renovating events for the processX,(???)

and thatXim 1Aj,m is Gj,mmeasurable, for all i jm. Then

:= inf{n 0 : 1An = 1}

is a verifiable time with respect to the {Gj,m}.

Proof. We shall show that= supm0 (m), for appropriately defined backwards G,mstoppingtimes(m)that satisfy the properties (i), (ii) and (iii) of Definition 4. Let

(m) := inf{j 0 : 1Aj,m = 1}.

SinceAj,m Gj,m, we immediately have that(m)is a backwards G,mstopping time. Then

(m) := inf{j0 : jB, . . . , m Bm+j}

is a.s. increasing inm, with

supm

(m) := inf{j0 : jB, . . .}= inf{j0 : 1Aj = 1}= .

Hence (i) of Def. 4 holds. We next use the fact that the An are renovating events. we have, for alli j ,

Xim 1Aj =Xjm 1Aj , a.s.

Since

Aj =Aj,m {m+1 Bm+1+j, . . .}=: Aj,m Dj,m,

we have

Xim 1Aj,m1Dj,m =Xjm 1Aj,m1Dj,m , a.s.

By assumption,Xim 1Aj,m is Gj,mmeasurable, for alli j . By the independence between thens,Dj,m is independent ofGj,m. Hence, by Lemma 1 of the Appendix, we can cancel the 1Dj,mterms in the above equation to get

Xim 1Aj,m =Xjm 1Aj,m, a.s.,

for alli j . Now,{(m) =j} Aj,m, (8)

and so, by multiplying by 1((m) =j)both sides, we obtain

Xim 1((m) =j) =Xjm 1((m) =j), a.s.,

for alli j . By stationarity,(m)< , a.s., and so for all 0,

X(m)m =X(m)m , a.s.

Hence (ii) of Def. 4 holds. Finally, to show thatX(m)m is G(m),mmeasurable, we show that

Xjm 1((m) =j)is Gj,mmeasurable. Using the inclusion (8) again, we write

Xjm 1((m) =j) =Xjm 1Aj,m1((m) =j).

By assumption,Xjm 1Aj,m is Gj,mmeasurable, and so is 1((m) = j). Hence (iii) of Def. 4

also holds.


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A perfect simulation algorithm

In the remaining of this section, we describe a perfect simulation algorithm, i.e., a method for

drawing samples from the stationary version of a process. The setup is as in Theorem 2. For

simplicity, we take = 0. That is, we assume that

A0= {0 B0, 1 B1, . . .}

has positive probability, and that the An = nA0 are renovating events for the process {Xn}.

Recall that the {n = 0n} are i.i.d., and that Gm,n = (m, . . . , n),m n. It was proved in

Theorem 2 that the time= inf{n 0 : 1An = 1} is a bc-time which is verifiable with respectto the{Gm,n}. This time is written as = supm0 (m), where (m) = inf{j 0 : j B0, . . . , mBm+j}. The algorithm uses(0)only. It is convenient to let

1:= (0) = inf{j0 : j B0, . . . , 0 Bj},

i+1:= i+ (0)i , i 1.

In addition to the above, we are going to assume that

B0 B1 B2 . . .

It is easy to see that this monotonicity assumption is responsible for the following

1j 1 j, a.s. on {1 j}. (9)

Owing to condition (ii) of Definition 4 we have

X10 =X0=X

1i0 , for anyi 0.

That is, if we start the process at time 1, we have, at time 0, that X0is a.s. equal to the stationary

X0. Applying j at this equality we have X10 j = X0j = Xj . But X10 j =(X1

1)j =X1j1

jj =Xj1j

j . That is,

Xj1j

j =Xj =X

j1ji

j , for anyi 0. (10)

But from (9), we have1 j + 1j , if1 j , and so, from (10),

X1j =Xj , a.s. on {1 j}.

This means that if we start the process at1, then its values on any window [j, 0] contained in[1, 0]match the values of its stationary version on the same window:

(X1j , . . . , X

10 ) = (Xj, . . . , X0), a.s. on {1j}. (11)

It remains to show a measurability property of the vector (11) that we are simulating. By (iii) of

Definition 4, we have that X10 is G1,0measurable. That is, if1 = thenX0 is a certain

deterministic function of, . . . , 0. Thus, the functionsh are defined, for all 0, by thecondition

X0 =h(, . . . , 0), a.s. on {1 = },

or,

X10 =h1(1 , . . . , 0).

Hence for anyi 0,

Xi1

i

i =X10 i =h1i(i1i , . . . , i).


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But if1 j , we have 1i 1i for all i [0, j], and so every component of(X

10

i, 0i j )is a deterministic function of0, . . . , 1 . Thus the vector appearing in (11) is a determin-istic function of0, . . . , 1 , if1j . This is precisely the measurability property we need.

We now observe that, in (11), we can replace 1 by anyi:

(Xij , . . . , X i0 ) = (Xj , . . . , X0), a.s. on {i j}, i= 1, 2, . . .

Hence if we want to simulate (Xj, . . . , X0) we search for an i such that i j, and start theprocess from i. It is now clear how to simulate the process on any window prior to 0.

To proceed forward, i.e., to simulate {Xn, n > 0}, consider first X1. Note that

X1= X0= h1(1, . . . , 0)

=h1(1+1, . . . , 1)

Next note that 1is either equal to0, or to 1+1, or to 2+1 =1+11+1, etc. This follows

from the definition of1 and i, as well as the monotonicity between theBj . If1= 0(which is to

say, 1 B0) then X1= h0(1). Otherwise, if1B0, but 1B1+1, then 1= 1+1, and soX1 =h1+1(1, . . . , 1). Thus, for some finite (but random)j (defined from1Bj+1\ Bj ),

we have X1= hj+1(j , . . . , 1). The algorithm proceeds similarly forn >1.

The connection between perfect simulation and backward coupling was first studied by Foss

and Tweedie (1998).

Weak verifiability

Suppose now that we drop the condition that P(A0)> 0, but only assume that

(0)< , a.s.

Of course, this implies that(m) < , a.s., for allm. Here we can no longer assert that we havesc-convergence to a stationary version, but we can only assert existence in the sense described in

the sequel. Indeed, simply the a.s. finiteness of(0) (and not of) makes the perfect simulationalgorithm described above realizable. The algorithm is shift-invariant, hence the process defined

by it is stationary. One may call this process a stationary version ofX. This becomes precise if{Xn} itself is a stochastic recursive sequence, in the sense that the stationary process defined by thealgorithm is also a stochastic recursive sequence with the same driver. (See Section 4.)

The construction of a stationary version, under the weaker hypothesis(0) < , a.s., is alsostudied by Comets et al. (2001), for a particular model. In that paper, it is shown that (0) < a.s., iff

n=1

nk=0

P(0 Bk) =.

The latter condition is clearly weaker than P(A0) > 0. In Cometset al. (2001) it is shown thatit is equivalent to the non-positive recurrence of a certain Markov chain, a realization which leads

directly to the proof of this condition.

4 Strong coupling for stochastic recursive sequences

As in the previous section, let(,F, P , )be a probability space with aP-preserving ergodic flow

. Let(X,BX),(Y,BY) be two measurable spaces. Let {n, n Z}be a stationary sequence


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ofY-valued random variables. Letf : X Y X be a measurable function. A stochasticrecursive sequence (SRS) {Xn, n 0} is defined as an X-valued process that satisfies

Xn+1= f(Xn, n), n 0. (12)

The pair(f, {n})is referred to as the driver of the SRS X. The choice of0 as the starting point isarbitrary.

Astationary solution {Xn} of the stochastic recursion is a stationary sequence that satisfies theabove recursion. Clearly, it can be assumed that Xn is defined for all n Z. There are examplesthat show that a stationary solution may exist but may not be unique. The classical such example is

that of a two-server queue, which satisfies the so-called Kiefer-Wolfowitz recursion (see Brandt et

al. (1990)). In this example, under natural stability conditions, there are infinitely many stationary

solutions, one of which is minimal and another maximal. One may define a particular solution,

say{X0n}, to the two-server queue SRS by starting from the zero initial condition. ThenX0 sc-

converges (under some conditions) to the minimal stationary solution. In our terminology then, we

may say that the minimal stationary solution is the stationary version ofX0.

Stochastic recursive sequences are ubiquitous in applied probability modeling. For instance, a

Markov chain with values in a countably generated measurable space can be expressed in the form

of SRS with i.i.d. drivers.

The previous notions of coupling take a simpler form when stochastic recursive sequences are

involved owing to the fact that if two SRS with the same driver agree at some n then they agreethereafter. We thus have the following modifications of the earlier theorems:

Proposition 3. LetX, Xbe SRS with the same driver(f, {n}), and assume thatX is stationary.Then

(i)Xc-converges to X ifflimn

P(Xn = Xn) = 1.

(ii)Xsc-converges to X iff

limn

P(Xn = Xn = X1n =X

2n = ) = 1.

(iii)Xbc-converges iff

limn

P(Xn0 =X(n+1)0 =X

(n+2)0 = ) = 1.

The standard renovation theory (see Borovkov (1984,1998)) is formulated as follows. First,

define renovation events:

Definition 5 (renovation event for SRS). Fix n Z, Z+and a measurable function g: Y+1 X. A setR Fis called(n,,g)renovating for the SRSX iff

Xn++1= g(n, . . . , n+), a.s. onR. (13)

An alternative terminology (Borovkov (1988)) is:R is a renovation event on the segment[n, n + ].

We then have the following theorem.

Theorem 3(renovation theorem for SRS). Fix 0 andg : Y+1 X. Suppose that, for eachn 0, there exists a(n,,g)renovating eventRn forX. Assume that{Rn, n 0}is stationaryand ergodic, with P(R0)> 0. Then the SRSXbc-converges and its stationary version Xis an SRS

with the same driver asX.


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Proof. For eachn Z, define Xn,i, recursively on the indexi, by

Xn,n++1= g(n, . . . , n+)

Xn,n+j+1= f(Xn,n+j, n+j), j + 1, (14)

and observe that (13) implies that

n 0 j + 1 Xn+j = Xn,n+j, a.s. onRn. (15)

Forn Z, setAn := Rn1. Consider the stationary background

Hn,p:= Xn1,p, p n.

Note thatHn,pk =Hn+k,p+k and rewrite (15) as

n + 1 i 0 Xn+i = Hn,n+i, a.s. onAn.

Since P(A0) = P(R0) > 0, there is a unique stationary version X, constructed by means of the

bc-time := inf{i 0 : 1Ri1 = 1}. (16)

We have

Xn = X+n =H,n = X1,n.

This Xncan be defined for all n Z. From this, and (14), we have that Xn+1= f(Xn, n), n Z,i.e., Xhas the same driver asX.

It is useful to observe that, ifRnare(n,,g)renovating events forXwith P(R0)> 0, then thestationary version X satisfies X = g (1, . . . , 1), a.s., whereis the bc-time definedin (16). More generally, if we consider the random set {j Z : 1Rj = 1}(the set of renovation

epochs), we have, for any in this set, X= g(1, . . . , 1), a.s.

Example 2. Consider 2-server queue with stationary ergodic driving sequences {(n, tn)}.

Example 3. Consider a Markov chain {Xn} with values in a finite set S, having stationary transitionprobabilities pi,j , i, j S. Assume that [pi,j ] is irreducible and aperiodic. Although there is aunique invariant probability measure, whether Xbc-converges to the stationary Markov chain Xdepends on the realization ofXon a particular probability space. We can achieve bc-convergencewith a verifiable bc-time if we realize X as follows: Consider a sequence of i.i.d. random mapsn :S S,n Z(and we writenn+1 to indicate composition). Represent eachn as a vectorn = (n(i), i S), withindependentcomponents such that

P(n(i) =j) =pi,j , i, j S.

Then the Markov chain is realized as an SRS by

Xn+1= n(Xn).

It is important to notice that the condition that the components ofnbe independent is not necessaryfor the Markovian property. It is only used as a means of constructing the process on a particular

probability space, so that backwards coupling takes place. Now define

= inf{n 0 : i, j S 0 n(i) =0 n(j)}.

It can be seen that, under our assumptions, is a bc-time for X, < , a.s., and is verifiable.This bc-time is the one used by Propp and Wilson (1996) in their perfect simulation method for

Markov chains. Indeed, the verifiability property ofallows recursive simulation of the random

variable0 (i)which (regardless ofi) has the stationary distribution.


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Example 4. Another interesting example is the process considered by Bremaud and Massoulie

(1994) that has a Markov-like property with random memory. Consider a process{Xn, n Z}with values in a Polish space Sand suppose that its transition kernel, defined by P(Xn |Xn1, Xn2, . . .) is time-homogeneous [a similar setup is considered in the paper by Comets etal. (2001)], i.e. that there exists a kernel : K B(K) [0, 1] such that P(Xn B |Xn1, Xn2, . . .) = ((Xn1, Xn2, . . .); B), B B(K). This represents the dynamics of theprocess. In addition, assume that the dynamics does not depend on the whole past, but on a finite

but random number of random variables from the past. It is also required that the random memory

is consistent and that the minorization condition ((Xn1, Xn2, . . .), ) (), where (0, 1) and a probability measure on (K,B(K)), holds. See Bremaud and Massoulie (1994) fordetails. Then it is shown that renovation events do exist and that the process {Xn} sc-converges toa stationary process that has the same dynamics.

Appendix: Auxiliary results

Lemma 1. IfY1, Y2 andZ are three random variables such thatZ is independent of(Y1, Y2),P(Z= 0)> 0, andY1Z=Y2Z, a.s., thenY1= Y2, a.s.

Proof. Since P(Y1Z=Y2Z) = 1, we have

P(Z= 0) = P(Y1Z=Y2Z, Z= 0) = P(Y1 = Y2, Z= 0) = P(Y1 = Y2)P(Z= 0),

where the last equality follows from independence. Since P(Z = 0) > 0, we obtain the resultP(Y1= Y2) = 1.

Proposition 4. LetX1, X2, . . . be a stationary-ergodic sequence of random variables with EX+1 0 then, for any0 < < c/2, there isk0 such that P(Yk/k > c+)< for allk k0. Fixk k0 and letn = 2k. We then have

P(Yn/n > 3c/4) P(Yn/n > (c + )/2)

P(max(Yk/n,Ykk/n)> (c + )/2)

2P(Yk/n > (c + )/2)

= 2P(Yk/k > c + ),

which contradicts the a.s. convergence ofYn/n to c. Hence c = 0. To show that EYn/n 0simply observe that the sequence Yn/n is bounded by Sn/n = (X1 + + Xn)/n, and sinceSn/n EX1, a.s. and inL

1, it follows that{Sn/n}is a uniformly integrable sequence, and thusso is {Yn/n}.

Proposition 5. LetX1, X2, . . .be a stationary-ergodic sequence of random variables with EX1=0. Consider the stationary walkSn= X1+ +Xn, n 1, with S0= 0. PutMn = max0in Si.

ThenMn/n 0, a.s. and inL1.


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Proof. Fix >0. LetM = supi0(Si i). ThenM


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[17] Lindvall, T. (1992, 2003)Lectures on the Coupling Method. Wiley, New York.

[18] Propp, J.G. and Wilson, D.B. (1996) Exact sampling with coupled Markov chains and appli-

cations to statistical mechanics.Random Structures and Algorithms9, 223-252.

[19] Thorisson, H. (2000)Coupling, Stationarity, and Regeneration.Springer-Verlag, New York.

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