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SECTION 36. KOLMOGOROV'S EXISTENCE THEOREM

Stochastic Processes

A stochastic process is a collection [X,: t l T] of random variables on aprobability space (2, Jr, P). The sequence of gambler's fortunes in Section7, the sequences of independent random variables in Section 22, thequeueing process of Section 24, the martingales of Section 35—all these arestochastic processes for which T = (1, 2, ... ). For the Poisson process [N,:t ^ 0] of Section 23, T = [0, cc). For all these processes the points of T arethought of as representing time. In most cases, T is the set of integers andtime is discrete, or else T is an interval of the line and time is continuous.For the general theory of this section, however, T can be quite arbitrary.

Finite-Dimensional Distributions

A process is usually described in terms of distributions it induces inEuclidean spaces. For each k-tuple (t 3 , ... , t k ) of distinct elements of T, therandom vector (X,,,..., X,k ) has over T k some distribution p,,,... 6 :

(36.1) C„."000..*J+"= PRX,,,..., X,k ) e JK. H Ka r .

These probability measures 12, m ... ,, are the finite-dimensional distributions ofthe stochastic process [X,: t l T]. The system of finite-dimensional distri-butions does not completely determine the properties of the process. Forexample, the Poisson process _P.<"t ^ 0] as defined by (23.5) has samplepaths (functions N,(w) with w fixed and t varying) that are step functions.But (23.28) defines a process which has the same finite-dimensional distribu-

SECTION 36. KOLMOGOROV'S EXISTENCE THEOREM 507Lions and has sample paths that are sty"step functions. Nevertheless, thefirst step in a general theory is to construct processes for given systems offinite-dimensional distributions.

Now (36.1) implies two consistency properties of the system D t."xk .Suppose the H in (36.1) has the form H = H1 X • • • x Hk (H, e gi p ), andconsider a permutation dr of (1,2, k). Since[(X, 6 , X,k ) E (H3x • • • x Kk and R , Xm0k ) E (141 X - - - Xiiwk )] are the same event,it follows by (36.1) that

(36.2) X • [Kk +"= X • • • [K„k +•

For example, if tt w,, = v x {)."then necessarily = v' x v.The second consistency condition is

(36.3) A x, ... 4_,(1/3 • • xHk _ m )

= -•- tk-lik(K1 ° • • • xHk_i x R').

This is clear because [..pd3+"lies in H3 x • xHk _ 3 if and only if(Xxm , , X,k -1, X,k ) lies in H3 X X Hk_ m X Rl.

These conditions can be stated in a slightly different way. Define cp.:Rp"R p"f°

=

hu0"applies the permutation 4u to the coordinates (for example, if IT sends x 5

to first position, then ir /31 = 3). Since cp; 3(H3 x x Kk +"= 143

x • • • xH„k , it follows from (36.2) that

134, ---1„„c1C 3 (H) = At, yk *K+

for rectangles H. But then

(36.4) ••• yp"? yjwpqSny"1

Similarly, if cp: U k"--) R k-1 is the projection hu*° i . ."°k +"?"*°l . ."°"p

then (36.3) is the same thing as

(36.5) Cnk!•!•!•!tk -2!>!Kv!#!•!2m)R!2(

508 UVQCJCUVKC!RRQCEUUEU

The conditions (36.4) and (36.5) have a common extension. Suppose that(u 2, , um ) is an m-tuple of distinct elements of T and that each elementof (t 2, t k ) is also an element of (u 2, ..., um ). Then (t 2, t k ) must bethe initial segment of some permutation of (u 2 , ..., um ); that is, k m andthere is a permutation ¶ of (1, 2, ... , m) such that

u„-1„,) = /— 1 3555",tk,tk+1,•••,tm),

where t k+1 , t„, are elements of (11 ,Op • • •k •t Define tp: /V" –0 R k by

, u m ) that do not appear in

(36.6) = x„-ik);

applies IT to the coordinates and then projects onto the first k of them.Since Ip( Xwk, . , Xu. ) = , X2m ),

(36.7) si 12 2 np"C 191 1.9.Z •

This contains (36.4) and (36.5) as special cases, but as 1w is a coordinatepermutation followed by a sequence of projections of the form (x 2, , x 3 )

(x m ,...,x m _ m ), it is also a consequence of these special cases.Measures C mp"mk"coming from a process via (36.1) necessarily satisfy

(36.2) and (36.3). The problem is to show conversely that, if (36.2) and(36.3) hold for a given system of measures, then there exists a processhaving these finite-dimensional distributions. Proving this theorem is themain objective of the section.

Product Spaces

The standard construction of the general process involves product spaces.Let T be an arbitrary index set and let R V be the collection of all realfunctions on T—all maps from T into the real line. If T = (1, 2, ... , k ), areal function on T can be identified with a k-tuple (x 2, , xm ) of realnumbers, and so R V can be identified with k-dimensional Euclidean spaceR mc. If T = (1, 2, ... ), a real function on T is a sequence (x 2 , x 2 , ... ) ofreal numbers. If T is an interval, R V consists of all real functions, howeverirregular, on the interval.

Whatever the set T may be, an element of R V will be denoted x. Thevalue of x at t will be denoted x(t) or x„ depending on whether x isviewed as a function of t with domain T or as a vector with componentsindexed by the elements t of T. Just as R k can be regarded as the cartesianproduct of k copies of the real line, R V can be regarded as a product space

VHFWLRQ"36. NROPRJRURY)V"H[LVWHQFH"WKHRUHP 509—a product of copies of the real line, one copy for each t in T.

For each t define a mapping Z1 : R V —0 R m by

(36.8) _.*°+"?"°*y+"= x,.

The Z, are called the coordinate functions or projections. When later on aprobability measure has been defined on UV."the Z, will be randomvariables, the coordinate variables. Frequently, the value _.*°+"is insteaddenoted _*y."x). If x is fixed, Z(•, x) is a real function on T and is, in fact,nothing other than °*"• )—that is, x itself. If t is fixed, _*y."• ) is a realfunction on UV and is identical with the function Z, defined by (36.8).

There is a natural generalization to UV of the idea of the a-field ofk-dimensional Borel sets. Let UV be the a-field generated by all thecoordinate functions Z„ t i"T: .9PV = a[Z,: t H"T]. It is generated by thesets of the form

_}"H"R V : Zx (x) i"KM"= [x i"R V : x x"i"111

for y"i"T and H E .02 2. If T = (1, 2, ..., k}, then .9i V coincides with R m.Consider the class .94 consisting of the sets of the form

(36.9) A = Ex € R V :(Z,i (x),...,Z,k (x)) E Km

?"m}"i"UV < i"313.

where k is an integer, (t 2 , yk +"is a k-tuple of distinct points of T, andH i"Pk0"Sets of this form, elements of .9-t'qV, are called finite-dimensional sets,or cylinders. Of course, #"generates .9r. Now RI' is not a a-field, doesnot coincide with MT (unless T is finite), but the following argument showsthat it is a field.

The complement of (36.9) is UV – A = [x H"UV <"(x,,,... , x, m ) H"U k –H], and so yPo is closed under complementation. Suppose that A is givenby (36.9) and B is given by

(36.10)

B = [x E RV : (x so - • • , 3a.

where I i"°SM0"Let (u2 , um ) be an m-tuple containing all the t q and allthe sp . Now (t 2, , yk +"must be the initial segment of some permutation of(u 2, , um ), and if 4 is as in (36.6) and H' = then H' H".9Pm and Ais given by

(36.11) A= [x i"UV <"*} ui . } um+"H']

510 STOCHASTIC PROCESSES

%2

a2

Sm

e 1

2If V"is an interval, the cylinder [x E RT:a, < x(ro ^ fii , a2 < x(t2 ) ^ $2 1 consists of thefunctions that go through the two gates shown; y lies in the cylinder and z does not (they neednot be continuous functions, of course).

as well as by (36.9). Similarly, B can be put in the form

(36.12) B = [x H"UV <"*}0..000."}00+"i"I'],

where I' i"Rm. But then

(36.13) A U B = [x e R V : (x ui , ..., Xu. ) €"H' U P].

Since H' U I' i"A', A U B is a cylinder. This proves that gif is a fieldsuch that gr 0 = a(.11).

The Z, are measurable functions on the measurable space (R T, gPT). If Pis a probability measure on AP T then [4: t i"T] is a stochastic process on(R r, 9117',"t+,"the coordinate-variable process.

Kolmogorov's Existence Theorem

The existence theorem can be stated two ways:

Theorem 36.1. If A !, ... ,, are a system of distributions satisfying the con-sistency conditions (36.2) and (36.3), then there is a probability measure P on.9117. such that the coordinate-variable process [Z 1 : t i"T] on (R T, .42T, P) hasthe 11,,... ,, as its finite-dimensional distributions.

Theorem 36.2. If N... ,, are a system of distributions satisfying the con-sistency conditions (36.2) and (36.3), then there exists on some probabilityspace (SI, .9', P) a stochastic process [X,: t i"T] having the 11, 4 ... ,,, as itsfinite-dimensional distributions.

SECTION 36. KOLMOGOROV'S EXISTENCE THEOREM 511For many purposes the underlying probability space is irrelevant, the

joint distributions of the variables in the process being all that matters, sothat the two theorems are equally useful. As a matter of fact, they areequivalent anyway. Obviously, the first implies the second. To prove theconverse, suppose that the process [X,: t E!T] on (0, ..F, P) has finite-dimensional distributions 11, 1 ...,k, and define a map i: S/ —0 R T by therequirement

(36.14)

Z,(i(o.))) = X3 (4.), t E T.

For each (4), i(6.) is an element of R T, a real function on T, and therequirement is that X,(w) be its value at t. Clearly,

(36.15)

knz!E!TT <*6 1 *}+.000.=*}++"E!H]

= ^yg!SI: (Z,1 (i(0),...,Z,,(06.0)) E!H]

= [to €!UK:!*Zli *{+.000."Ztk *6+++"g!Jm=

since the X, are random variables, measurable F, this set lies in F ifH E!e k 0"Thus i'A E!H!for A E!RI; and so (Theorem 13.1) t is measur-able .f/.9PW. By (36.15) and the fact that [X,: t E!T] has finite-dimensionaldistributions A nn ...,,, Pf -1 (see (13.7)) satisfies

(36.16) Pi - 1[x E!TT <*]ti *}+.000.=*}++"E!H]

= P[w E!U]:!(Xii (w),...,X,k(w)) E!H] = p,,, ...,,(H).

Thus the coordinate-variable process [Zr : t g!T] on (RT, RT, Pr') alsohas finite-dimensional distributions 12,, ..., k.

Thus to prove either of the two versions of Kolmogorov's existencetheorem above is to prove the other one as well.

Example 36.1. Suppose that T is finite, say T = (1, 2, ... , k). Then(RT."UW+"nw"*Rk . Rk +. and taking p = 1.

1 1,2,..., p"satisfies the requirements ofTheorem 36.1. ■

Example 36.2. Suppose that T = (1,2, ... ) and

(36.17) I"11 • • • fk = µl1 ` • • • `"P."le

622 UVQCJCUVKC!RRQCEUUEU

where A l , ta, 4 , ... are distributions on the line. The consistency conditionsare easily checked. By Theorem 20.4 there exists on some (0, , P) anindependent sequence X3 , X2-///!of random variables with respective distri-butions A i, /!But then (36.17) is the distribution of ( X11, , X1k ). Forthe special case (36.17), Theorem 36.2 was thus proved in Section 20. Theexistence of independent sequences with prescribed distributions was, infact, the measure-theoretic basis of all the probabilistic developments ofChapters 4, 5, and 6—even dependent processes like the Poisson andqueueing processes were constructed from independent sequences. Theexistence of independent sequences can also be made the basis of a proof ofTheorem 36.2 in its full generality; see the second proof below. ■

Example 363. If T is a subset of the line, it is convenient to use theorder structure of T and take the 12,1 ,k to be specified initially only fork-tuples (t 3, , t k ) that are in increasing order:

(36.18) tl < t4 < • • • < t k .

It is natural for example to specify the finite-dimensional distributions forthe Poisson process for increasing t 3 , , t k alone; see (23.27).

Assume that the ,k for k-tuples satisfying (36.18) have the con-sistency property

(36.19)•••vk-kvk,k(••!vm *333"[!•!•!• z!Ji+1!z!•!•!•!X24*

> X!•!•!• X!R L X Hi+ , X • • • XHk ).

If (36.18) does not hold, let ,r be the permutation for which

(36.20) < t,,4 < <

and define by the right side of (36.2) (or (36.4)). This defines thefinite-dimensional distributions for all k-tuples of distinct points of T, andthey satisfy the two consistency conditions of Kolmogorov's theorem. To seethis, consider first a permutation p of (1, 2, ... , k), and put s. = ttm and

= Hai. If w is determined by (36.20), then p / kr is the permutation thatputs the s m in increasing order, because st -1,93 = 41 . But then

11,2 ... .Sk*Kws x • • xHtk ) = sk(J1 x - xJk )

= p, . . . „04 ( Jt - x • • • x Irk )

= yt tia,(141 x • • • X H£wp )

= tk(H1 X • • • X Hk),

SECTION 36. KOLMOGOROV'S EXISTENCE THEOREM 513where the second and the fourth equalities hold because of the way thefinite-dimensional distributions have been defined for the general k-tuple.The ik as thus extended therefore satisfy (36.2), the first of theconsistency conditions.

Suppose of the IT determined by (36.20) that k = irj. By definition,

(36.21) X • xHk_l X le)

= µ yGs vj.k)!Hei X!///!a"Rm!X!•!•!•!X!Hek *-

where on the right le is the jth component of the product. By theassumption (36.19), this le can be canceled if the index t £wj = t p is canceledin the subscript to the 11; since t mr , , c(i+1), t„p are t 3 , ,in increasing order, the definition further reduces the right side of (36.21) to

(Hs • • • x Hk _ K). The D rc .. xk thus satisfy (36.3), the secondconsistency condition.

It will therefore follow from the existence theorem that if T c le andx, is defined for all k-tuples in increasing order, and if (36.19) holds,

then there exists a stochastic process [X„ t e T] satisfying (36.1) forincreasing t m, , t p. ■

Two proofs of Kolmogorov's existence theorem will be given. The first isbased on the extension theorem of Section 3.FIRST PROOF OF KOLMOGOROV'S THEOREM. Consider the first formulation,Theorem 36.1. If A is the cylinder (36.9), define

(36.22) P(A) =

This gives rise to the question of consistency because A will have otherrepresentations as a cylinder. Suppose, in fact, that A coincides with thecylinder B defined by (36.10). As observed before, if (u m, , u,,,) containsall the t t and ss , A is also given by (36.11), where H' = 21H and tp isdefined in (36.6). Since the consistency conditions (36.2) and (36.3) implythe more general one (36.7), P(A)= Similarly,(36.10) has the form (36.12), and P(B) = u„(P). Sincethe u y are distinct, for any real numbers z 3, , z„, there are points x of RTfor which (x„,, , = (z 3 ...,z,„). From this it follows that if thecylinders (36.11) and (36.12) coincide, then H' = I'. Hence A = B impliesthat P(A) = u,... u,„( 33 ') = u„,(1')= P(B`), and the definition (36.22)is indeed consistent.

734"STOCHASTIC PROCESSES

Now consider disjoint cylinders A and B. As usual, the index sets may betaken identical. Assume then that A is given by (36.11) and B by (36.12), sothat (36.13) holds. If H' n I' were nonempty, then A r"B would benonempty as well. Therefore, H' r"I' = 0, and

P(A U B) = U I')

= u.,(10 + um(P)= P(A) + P(B).

Therefore, P is finitely additive on lnmk0"Clearly, P(R V) = 1.Suppose that P can be shown to be countably additive on UgW0"By

Theorem 3.1, P will then extend to a probability measure on .9r. By theway P was defined on TK=

P[x H"R V :(;(x), , 4k(x)) E H] = 4 ... ik (H),

and therefore the coordinate process _6<"x"H"T] will have the requiredfinite-dimensional distributions.

It suffices, then, to prove P countably additive on .9{(Z1 and this willfollow if A n"i"9/1. and A n 4, 0 together imply P(A,,) 91 0 (see Example 2.10).Suppose that A p 0 A 4"0 -• • and that P(A„) >— E> 0 for all n. Theproblem is to show that r„C n must be nonempty. Since A n"H"ku."and sincethe index set involved in the specification of a cylinder can always bepermuted and expanded, there exists a sequence t 3 , t 4 , ... of points in T forwhich

C„"= [x i"UV<"i"Kr a.

wheret Hn"H".91".Of course, P(A„)= tt ii ...,,n(Hn ). By Theorem 12.3 (regularity), there

exists inside Hn a compact set K,, such that K,,)< c/2n+1.If

B,,= [x H"UW<"*} mm .000."° x0 +"i"K,,], then P(A n — BO< E/2"1. Put E„"=r.rcc.Ek 0"Then E„"c Bn C A n and P(A„ — CO< c/2, so that P(Cn )> c/2> 0. Therefore, E„"is nonempty.

Choose a point x ( n ) of R V in E„0"If n > k, then x ( n )"i"Fr"F"Ck C Bkand hence *}.*3$ +.", 6<3+ +"E Kk. Since Kk is bounded, the sequence{ x (1) x(2)

•"•"•"} is bounded for each k. By the diagonal method [A14] select

Lp"Lpan increasing sequence n m , n 4 , of integers such that lim mxV exists for

*In general, C„"will involve indices t,, t2 ,... , xon."where a n < a 2 < • • • . For notationalsimplicity a„ is taken as s="fx"f"matter of fact, this can be arranged: start off by repeating 7el — 1 times and then for s"3"wjujfy"C„"en+1 — a„ times.

SECTION 36. KOLMOGOROV'S EXISTENCE THEOREM 515

each k0"There is in TV"some point x whose t kth coordinate is this limit foreach k0"But then, for each k."*} }".k +"mw"xli"pmqmx"ew"m"—A"22"of

, .4:32 ) and hence lies in Kk. But that means that }"itself lies in B kand hence in A k . Thus x E ncko ,A u , which completes the proof t ■

The second proof of Kolmogorov's theorem goes in two stages, first for countableV."then for general T.*SECOND PROOF FOR COUNTABLE T. The result for countable T will be proved in itssecond formulation, Theorem 36.2. It is no restriction to enumerate T as ( t3 , t4 , )and then to identify xo with r="in other words, it is no restriction to assume thatT= (1, 2, ... }. Write in place of IL 1.2.....n.

By Theorem 20.4 there exists on a probability space (52, ."R+"(which can betaken to be the unit interval) an independent sequence U1, U2, ... of randomvariables each uniformly distributed over (0,1). Let F3 be the distribution functioncorresponding to A p . If the "inverse" k1 of F3 is defined over (0,1) by kl *w+"=inf[ }<"w"7"Fm (x)], then X3 = g3 (U3 ) has distribution Cm by the usual argument:R_k1 *W1 +"}a"?"R_3031"w"H3*}+a"?"Fm (x)•

The problem is to construct X2, X3 , ... inductively in such a way that

(36.23)

Xk = hk(1.31,• • • . L3k)

for a Borel function l k and ( X3 , , Zr +"has the distribution Cr 0"Assume thatX3, , Z„c 1 have been defined *r"2): they have joint distribution and(36.23) holds for k 5 r"—"1. The idea now is to construct an appropriate conditionaldistribution function H„*}3}1 .000"."}„c i +="here Hr"*}"I Xm ( w), , X„_ 3 ( w )) will havethe value R_ 9.(1 1 1 Ayuf p ]. would have if Z„"were already defined. Ifkr*)"Ix ) , , }„c 1 +"is the "inverse" function, then Z„* = k„*36*8++3Z1 *gs+.

.Z„c 1 *s0;+"will by the usual argument have the right conditional distributiongiven X3, , Z„"_ 3 , so that ( X3, , xu f 1 1 xu . will have the right distribution over

To construct the conditional distribution function, apply Theorem 33.3 in(R", ;x$."r„+"xs"kix"a conditional distribution of the last coordinate of (x 3 , , }r +given the first n — 1 of them. This will have (Theorem 20.1) the formv*J="}1 .000"."}„"c 1 +="mx"is a probability measure as J"varies over 91' 1 , and

= µ„_}"G"Tr<*}„000.}n c l +"G"O."}n

Since the integrand involves only x p , , xr_ 3 and Cs by consistency projects to

t The last part of the argument is, in effect, the proof that a countable product of compact setsis compact.* This second proof, which may be omitted, uses the conditional probability theory of Section33.

738 UVQCJCUVKC!RRQCEUUEU

Cn c 1 under the map (x 2 , ... , x„) -, (x2 , ... , x„_ K ), a change of variable gives

jmz*J="}l .000.}„c 1 +"6„c 1 *}i .000.}nc i +

?"C n _}"E T$<*} 1 .000.}„c 1 +"i"O."}„"E Ja0

Define Hn *}m}l ."000"."}„c 1 +"?"z**/"ss."}m="} i ."000"."}„c 1 +0"Thenxn-1) is a probability distribution function over the line, F" (x Ifunction over T$ /3 ."and

Hr*•"K})."/"/".•"+"mw"a Borel

m F„(xix n , ...1 x„f 1 . ch.t„_ 3 ( xp 1 ...1 x„f p .M

?"A n [x H"U$<*}k .000.}„c 2 +"i"M,x„ ^"}a0

Put k„*yp} l ."000"."}„c 1 +?"mrjm}<"y"5 F„(xlx 2 ,...,x„_ 2 )] for 0 < u < 1. SinceHn *}p}1 .000.}„c 1 +"is nondecreasing and right-continuous in x, k„*yp} 1 .000.}„c 1 +^ x if and only if u 5 F„(xlx k ,.. 0"."}„c 1 +0"Uix"Z„"//"kr*WrpCvx."•"•"•"."Z„c3+•"Umrgi( X2 , ... , Z„c 1 +"has distribution p„_ 2 and by (36.23) is independent of U,,, anapplication of (20.30) gives

P[( Xm , . . . , Xn _ 3 ) E iv, x„ ^ x]

= PRX19-1Xn-1) i Al . Un ^ Fn(XIX19— . Xn)]

C mm P`U„"^ In ( x 'xi 1 ... 1 x„f 1 .d hLyyn d 8 *"xi 1.. . 1 x„f i .

= m Fn (xpxi 1...1 xnf i . k{{uf 1 ( x i 1 ...1 x„f i .M

= ',ir k E T$<*} l .000.}„c 1 +"i"O."}n"^"}a0

Thus ( X2 , ... , Zn +"has distribution iL„. Note that Zn ."as a function of X2 , ... , Z„c 1and W.„"is a function of U2 , ... , u„ because (36.23) was assumed to hold for k">"r0Hence (36.23) holds for k"?"r"as well. ■

SECOND PROOF FOR GENERAL T. Consider (RV, RT) once again. If S c V."pix00,<s"?"e_].<"x"€"Ua0"Then Swu c .9rV = 9r.

Suppose that U"is countable. By the case just treated, there exists a process _Z.<x"E Ua"on some (SI, .F, R+—xli"process depends on U—wygl"that (X,k ,..., A%) hasdistribution 304"f!2"0"0"0"-m for every k-tuple (t 2 ,... , tk ) from U0"Define a map E: SI --) TVby requiring that

zt(i(6.)) = X,( w) if t € s,0 if t e S.

SECTION 36. KOLMOGOROV'S EXISTENCE THEOREM 517Now (36.15) holds as before if t 3 , , tk all lie in S, and so E is measurable .,"/"-w .Further, (36.16) holds for ts , tk in S. Put Rs = PE / 3 on 3w . Then Rs is aprobability measure on (RT,..f:7 ), and

(36.24) Ps[x ].m*}+.00"• , ].„*}++"e H] = ,k(H)

if H e 9i9k and ts ... , xk all lie in S. (The various spaces (0, R+"and processes[ x"E Ua"now become irrelevant.)

If Se c S, and if A is a cylinder (36.9) for which the ts , tk lie in Ss , thenRso *C") and Pw (A) coincide, their common value being ,k(H). Since thesecylinders generate Srws, Pws(A) Pw (A) for all A in ..Fws . If A lies both in .Fwm and.OrV4) Pthen Pwm (A) = Ps2(A). Thus P(A) —"Pw (A) consistently defines a- u w2 *C") =set function on the class Ws7vs ."the union extending over the countable subsets S ofT. If A„ lies in this union and A„e5rw0 (S„ countable), then S = U„ S„ iscountable and U„ A„ lies in 00Qvs 0"Thus Ws 00Hs is a a-field and so must coincide with9r. Therefore, P is a probability measure on 9r, and by (36.24) the coordinateprocess has under P the required finite-dimensional distributions. ■

The Inadequacy of glIT

Theorem 36.3. Let [X,: t e T] be a family of real functions on S2.(i) If A e a[X,: t e T] and w e A, and if Xt (w)= Xi(a) for all

t e T, then col e A.(ii) If A e a[ t E T], then A e a[ t E S] for some countable

subset S of T.

PROOF. Define t: S2 —• R T by Z,(t(w)) = X3(w). Let F= a[ X,: t E T].By (36.15), is measurable 9/913W and hence F contains the class [t / 1M:M E 9r]. The latter class is a a-field, however, and by (36.15) it containsthe sets [co E SZ : ;(a)) E H], H H"9P p."and hence containsthe a-field F they generate. Therefore

(36.25) a[X,: t e T] = M e

This is an infinite-dimensional analogue of Theorem 20.1(i).As for (i), the hypotheses imply that we A = t -1111 and i(co) = t(e),

so that w' E A certainly follows.For S c T, let Fw = a[ X,: t e S]; (ii) says that F= 007T coincides with= UwFw, the union extending over the countable subsets S of T. If

A 3 , C 2 ."000"lie in g, A„ lies in Fw0 for some countable S„, and so U„A„ liesin (9 because it lies in Fw for S = U„S„. Thus (9 is a a-field, and since itcontains the sets [X, H"H], it contains the a-field F they generate. (Thispart of the argument was used in the second proof of the existencetheorem.) ■

73:"UVQCJCUVKC!RRQCEUUEU

From this theorem it follows that various important sets lie outside theclass RT. Suppose that T = [0, oo). Of obvious interest is the subset C of RTconsisting of the functions continuous over [0, oo). But C is not in R V. Forsuppose it were. By part (ii) of the theorem (let SI = R W and put [Z,: t E!T]in the role of [X,: t g!T]), C would lie in a[Z,: t g!S] for some countableS c [0, oo). But then by part (i) of the theorem (let SZ = R V and put [Z,:t E S] in the role of [X,: t g!T]), if x g!C and Z,(x) = Z,(y) for allt K S, then y E!C. From the assumption that C lies in M W thus follows theexistence of a countable set S such that, if x E C and x(t) = y(t) for all tin S, then y E C. But whatever countable set S may be, for everycontinuous x there obviously exist functions y which have discontinuitiesbut which agree with x on S. Therefore, C cannot lie in RT.

What the argument shows is this: A set A in R V cannot lie in hLW unlessthere exists a countable subset S of T with the property that, if x E!A andx(t) = y(t) for all t in S, then y g!A. Thus A cannot lie in gr if iteffectively involves all the points t in the sense that, for each x in A andeach t in T, it is possible to move x out of A by changing its value at talone. And C is such a set. For another, consider the set of functions x overT = [0, oo) that are nondecreasing and assume as values x(t) only nonnega-tive integers:

)47/27*!^z!E!ng• 2 :!x(s) u!x(t), x"^"y="°*y+"g!)1-2-///!*-!t ^!1_/

This, too, lies outside gr.In Section 23 the Poisson process was defined as follows: Let X1 , [4."0"0"0

be independent and identically distributed with the exponential distribution(the probability space 0 on which they are defined may by Theorem 20.4 betaken to be the unit interval with Lebesgue measure). Put So = 0 andS„ = X1 + • • • +Xn . If S„(w) < Sn+1(41) ^for n 0 and S„( w) —> oo, putN(t, w) = N,(w) = max[n: S„( w ) ^ t] for t ^ 0; otherwise, put N(t, w) =N,(w) = 0 for t ^ 0. Then the stochastic process [N,: t ^ 0] has the finite-dimensional distributions described by the equations (23.27). The functionN(• , w) is the path function or sample functiont corresponding to w, and bythe construction every path function lies in the set (36.26). This is a goodthing if the process is to be a model for, say, calls arriving at a telephoneexchange: The sample path represents the history of the calls, its value at tbeing the number of arrivals up to time t, and so it ought to be nondecreas-ing and integer-valued.

According to Theorem 36.1, there exists a measure P on R V"jsv"T = _2."gs+such that the coordinate process [Z,: t > 0] on (R W, M V, P) has the finite-

v!Qvjgt!vgtmu!atg!realization qf!vjg!rtqeguu!apf!trajectory.

SECTION 36. KOLMOGOROV'S EXISTENCE THEOREM 62:

dimensional distributions of the Poisson process. This time does the pathfunction Z( • , x) lie in the set (36.26) with probability 1? Since Z( • , x) isjust x itself, the question is whether the set (36.26) has P-measure 1. Butthis set does not lie in .9iP V, and so it has no measure at all.

An application of Theorem 36.1 will always yield a stochastic processwith prescribed finite-dimensional distributions, but the process may lackcertain path-function properties which it is reasonable to require of it as amodel for some natural phenomenon. The special construction of Section 23gets around this difficulty for the Poisson process, and in the next section aspecial construction will yield a model for Brownian motion with continu-ous paths. Section 38 treats a general method for producing stochasticprocesses that have prescribed finite-dimensional distributions and at thesame time have path functions with desirable regularity properties.

RRQDNEOU

47/2/ 13.61 Generalize Theorem 20.1(ii), replacing (X3 , , Xk ) by [ X,: t e T]for an arbitrary T.

47/2/ A process ( , X _ 3 , X2 , X3 , .) (here T is the set of all integers) isstationary if the distribution of (Xn , A'u0 11 1 k _ m ) over le is the samefor all n = 0, ± 1, ± 2, .... Define r : R V"—> R V by 4, (Tx) = Z„+1(x); thusT moves a doubly infinite sequence (that is, an element of R V) one placeleft: T( , x_ 3 , x2 , x3 ,.. .) = (. . . , x2 , xp , x4 , . . .). Show that T is measur-able gi'V/R V and show that the coordinate process ( , Z_ 1 , )on (R V, 91+V, P) is stationary if and only if T preserves the measure P in thesense that PT -2 = P.

474/!Show that, if X is measurable a[ t e T], then X is measurable a[t e S] for some countable subset S of T.

47/4/!Suppose that [ X,: t E T] is a stochastic process on ( CI, ..F, P) and A EShow that there is a countable subset S of T for which P[AllX„ t E T] =P[AiiX t E S] with probability 1. Replace A by a random variable andprove a similar result.

47/6/!Let T be arbitrary and let K(s, t) be a real function over T X T. Supposethat K is symmetric in the sense that K(s, t) = K(t, s) and nonnegativedefinite in the sense that V-!K(ts 0 for k 1, t 3 , ... , t k in T,x 3 ,. , xk real. Show that there exists a process [X,: t E T] for which( , Xmk ) has the centered normal distribution with covariances K(t,,tm ),

j = 1, . , k.47/7/!9/4!V!Suppose that pr (u 3 , , u„) is a nonnegative real for each n and

each n-long sequence ut ... , u„ of 0's and l's. Suppose1 that p m (0) + p ma)

1 and pr-m (u3 ,.. ,u„,0) Rr-m*ym.0 ,u„,1) = Pr ui,. , ur ). Rtqxgvjav!qp!vjg!a-fkgnf!&(!igpgtavgf!b{!vjg!e{nkpfgtu!kp!ugswgpeg!uraeg!vjgtg