Random Semicontinuos Functions

Working Paper

RANDOM SEHlCONTINUOUS IWNCTIONS

Gabriella ~ a l i n e t t i ' Roger J-B. w e t s m m

August 1986 WP-86-47

International Institute for Applied Systems Analysis

A-2361 Laxenburg, Austria

NOT FOR QUOTATION

WITHOUT THE PERMISSION OF THE AUTHORS

RANDOM SEXICONTINUOUS FUNCTIONS

G a b r i e l l a ~ a l i n e t t i ' Roger J-B. w e t s a *

A u g u s t 1986 UT-86-47

Working P a p e r s a r e interim r e p o r t s on work of t h e I n t e r n a t i o n a l I n s t i t u t e f o r Applied System Analysis a n d h a v e r e c e i v e d only Limited review. Views o r opinions e x p r e s s e d h e r e i n d o n o t n e c e s s a r i l y r e p r e s e n t t h o s e of t h e Institute or of i t s National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 L a x e n b u r g , Austria

The s t u d y of dynamic s t o c h a s t i c optimization problems i s o f t e n hampered by a number of t e c h n i c a l complexities due t o t h e "classical" mathematical framework f o r S t o c h a s t i c P r o c e s s e s . Conceptually, as w e l l as technically, t h e c l a s s i c a l set-up is i n a p p r o p r i a t e f o r studying infima, allowing f o r approximations, e t c . H e r e t h e a u t h o r s i n t r o d u c e a n a l t e r n a t i v e a p p r o a c h which smooths o u t most of t h e s e diffi- c u l t i e s a n d g i v e s t h e study of s t o c h a s t i c p r o c e s s e s , in p a r t i c u l a r t h e s t u d y of func- tions of s t o c h a s t i c p r o c e s s e s , a n o t h e r p e r s p e c t i v e . This work s e r v e s as back- g r o u n d to IIASA's e f f o r t s in developing algorithmic p r o c e d u r e s f o r s t o c h a s t i c p r o - gramming and s t o c h a s t i c c o n t r o l problems.

Alexander B

.

K u n h a n s k i Chairman System a n d Decision S c i e n c e s P r o g r a m

CONTENTS

1 Stochastic Prooesses: The Classical View 2 Some Questions, Some Examples

3 Some Topological Considerations

4 Separability , Measurability and Stochastic Equivalence 5 The Epigraphical Approach

6 The Epigraphical Random S e t

7 Distributions and Distrfbution Functions 8

. . .

And Finite Dimensional Distribution!

9 Weak Convergence and Convergence in Distribution

10 Convergence in Distribution and Convergence of t h e Finite Dimensional Distributions

11 Bounded Random L.SC. Functions

12 An Application

to

Goodness-Of-Fit Statistics References

RANDOM SEXICONTINUOUS RJNCI'IONS

GabrieLLa ~ a l i n e t t i ' a n d Roger J-B. wets"

These notes introduce a new a p p r o a c h f o r t h e description, and t h e analysis.of stochastic phenomena. I t p a r t s company with t h e classical a p p r o a c h when t h e real- izations a r e infinite dimensional in nature. We shall b e mostly concerned with ques- tions of convergence and t h e description of t h e probability distributions associat- ed to such phenomena.

W e begin with a brief review of t h e classical t h e o r y f o r s t o c h a s t i c processes, bringing t o t h e f o r e some of t h e shortcomings of such a n a p p r o a c h . In t h e second p a r t of t h e p a p e r w e deal with t h e epigraphical a p p r o a c h t h a t r e l i e s on t h e model- ing of t h e "paths" of t h e s t o c h a s t i c phenomena by semicontInuous functions. W e conclude with a discussion a n d a comparison of t h e two theories, and t h e applica- tion t o t h e convergence of s t o c h a s t i c processes.

1.

STOCHASTIC PROCESSES: THE CLASSICAL VIEW

A stochcrstic process, with values in t h e extended r e a l s , Is a collection [Xt,

t

^E

Tj,

of extended real-valued random v a r i a b l e s indexed by T and defined on a probability s p a c e (Q, A, p). Here, and in t h e n e x t f e w sections, w e t a k e T

to

be a subset of

R.

I t i s a discrete process, if T i s a d i s c r e t e s u b s e t of

R,

in which c a s e , without loss of generality w e c a n always identify T with

Z

(the i n t e g e r s ) or

N

(the n a t u r a l numbers).

The probability m e a s u r e associated t o [Xt,

t

E

Tj

i s usually defined in terms of i t s finite dimensional distributions. For any finite s u b s e t Itl,

. . .

^,

tqj

C T, t h e q- dimensional random v e c t o r

defined on

(n,

A, p ) with values in

@ =

[- ^w,

-1s

h a s t h e probability measure de-

*UnivereltB di Roma

-

La Sapienza, 1-00185 Roma, Italy.

**University of California, Davis. California 95616, USA.

fined b y t h e c o r r e s p o n d e n c e

w h e r e

B

^E

gq

is a Bore1 s u b s e t of

@.

T h e family of p r o b a b i l i t y measures

w h e r e I(T) i s t h e collection of all f i n i t e s u b s e t s of T, i s t h e family of f i n i t e dimen- sional d i s t r i b u t i o n s of t h e s t o c h a s t i c p r o c e s s [X ,,

t

^E T j.

This a p p r o a c h i s a t t r a c t i v e for a n u m b e r of r e a s o n s , in p a r t i c u l a r b e c a u s e of its immediate simplicity,

at

l e a s t as f a r as t h e definition i s c o n c e r n e d . But in many cases, t h e p r i c e must b e paid a t a later stage, a n d sometimes t h e r e are t e c h n i c a l , a n d e v e n c o n c e p t u a l , difficulties t h a t c a n b e d i r e c t l y t r a c e d b a c k to t h i s "finite di- mensional" a p p r o a c h

to

s t o c h a s t i c p r o c e s s e s .

In a functional s e t t i n g , t h e c l a s s i c a l a p p r o a c h l e a d s

to

t h e following f r a m e - work. To e v e r y o E O, t h e r e c o r r e s p o n d s a f u n c t i o n (sample path, r e a l i z a t i o n ) :

The s t o c h a s t i c p r o c e s s IX,,

t

^E

Tj

c a n b e viewed as map from R i n t o

El';

we now identify

ET

with t h e s p a c e of all e x t e n d e d real-valued functions d e f i n e d o n

T.

The family of f i n i t e dimensional d i s t r i b u t i o n s a s s i g n s a p r o b a b i l i t y

to

all s u b s e t s of t h e t y p e

w h e r e B E

gq.

I

= Itl, . . .

,

t q j

E I ( T ) The sets B1 a r e c y l i n d e r s (with f i n i t e dimen- s i o n a l b a s e ) a n d t h e y form a field o n

3.

The f i n i t e dimensional d i s t r i b u t i o n s a s s i g n a m e a s u r e

to

e a c h set of t h i s field t h r o u g h t h e i d e n t i t y .

I t c a n b e shown, as d o n e by Daniel] a n d Kolmogorov, t h a t t h i s m e a s u r e P c a n b e uniquely e x t e n d e d

to

t h e o-field, d e n o t e d b y

F ,

g e n e r a t e d o n

El'

b y t h e family of c y l i n d e r s . W e c a n t h u s pin down a unique p r o b a b i l i t y m e a s u r e a s s o c i a t e d to t h e s b c h a s t i c p r o c e s s {X,, t ^E

Tj.

From t h i s viewpoint, t w o s t o c h a s t i c p r o c e s s e s are t h e n equivalent if t h e y h a v e t h e same f i n i t e dimensional d i s t r i b u t i o n s , t h e y i d e n t i f y t h e s a m e p r o b a b i l i t y m e a s u r e o n

3.

2.

SOME QUESTIONS. SOME EXAMPLES

One of t h e shortcomings of t h i s a p p r o a c h i s t h a t no attention i s paid

to

(possi- ble) topological p r o p e r t i e s of t h e realizations of t h e process. In many applica- tions, w e may b e i n t e r e s t e d in developing a calculus f o r p r o c e s s e s t h a t have v e r y specific p r o p e r t i e s , whose p a t h s may v e r y w e l l belong to a s u b s e t of of measure zero. The two following examples illustrate many of t h e difficulties.

EXAMPLE 2.1 Suppose V : fl --.(0, -) i s a random v a r i a b l e with continuous distribu- tion function. F o r a l l t E I& p r o b

CV = t ]

= O . Let T = R + a n d lYt, t ~ T j , f Y i ,

t

E T ~ be two stochastic p r o c e s s e s such t h a t

e x c e p t t h a t : Yt(o)

=-

¹ if V(o)

= t

These two p r o c e s s e s are equivalent, although t h e realizations of {Y; a r e continu- ous with probability 1, and those of f Y t j are continuous with probability 0.

One may be templed

to

view t h e phenomena illustrated by Example 2.1 as just a n o t h e r example of t h e f a c t t h a t random variables t h a t have t h e same distribution are not necessarily almost s u r e l y equal. But in t h i s case t h e r e i s something more t h a t e n t e r s i n t o play. Let C(T) denote t h e set of continuous frcnctions defined on T, and values in

E.

Thus, w e could reformulate o u r earlier observation, in t h e follow- ing terms:

pry:

EC(T)I = 1

,

and

pry.

EC(T)l = 0

,

but, as we shall now s e e , n e i t h e r C(T) n o r i t s complement

-

t h e s p a c e of functions with discontinuities

-

^belong

to

RT. The preceding expressions make s e n s e only be- cause

lo

^E n ( y r (0 ) E c(T)j

=

fl E A and

lo

^{E fllY.(o)} E c ( T ) ~

=

fl € A

.

But in terms of t h e probability distributions P and P' on R T induced by f Y t j and

[ Y i ^j

respectively, t h e expressions Pr(C(T)) =1, and P(C(T))

=

0 do not make sense because n e i t h e r P' n o r P are defined f o r t h e set C(T). To see this, simply ob- s e r v e t h a t since P

=

P', t h e above would imply

1

=

P (ET)

=

P(C(T)

u ⁽³

^{\ C (T))}

Observe also t h a t t h e p a t h s of both processes are bounded. But again in terms of P, o r equivalently I", w e cannot c h a r a c t e r i z e boundedness since

[x E IZTIO

s

x ( t )

s

1 , f o r a l l

t

^E

TI

g R~

.

EXAMPLE 2.2 m e Poisson Process. Let [X,. n

=

1 ,

...

] be a stochastic p r o c e s s (T

= N)

where

X l i s t h e waiting time f o r t h e f i r s t event, and f o r n

=

2,

...

X n i s t h e waiting time between (n

-

1)-th and n-th event.

Then, t h e time of o c c u r r e n c e of t h e n-th event i s

Under t h e assumption t h a t t h e event

O = : S , < S 1 < .-• < S n <

. . .

^{, Sup Sn n}

= -

^(2.1)

has probability 1, on t h i s s u b s e t of Q, w e define t h e random variables

t h a t r e c o r d s t h e (random) number of events t h a t o c c u r in t h e interval 10, t]; if o i s not in t h e set specified by (2.1), we set Nt(o) :

=

0. I t i s w e l l known, see [3] f o r ex- ample, t h a t if t h e IX,, n

=

^1,.

..

^{] are} independent with t h e same exponential distri- bution, then [Nt,

t r

0 ] i s t h e Poisson stochastic process.

For e v e r y o , t h e realization

i s a nondecreasing, integer-valued function.

Let

Q

C

R

be t h e rationals, and l e t q: [0, w) ⁴ Q be such t h a t q ( t ) :

= t

if

t

E

Q,

and q ( t ) : =O otherwise. Now, define

F o r a l l t E[O, -)

since Q

- t

i s a countable s u b s e t of R+ and

X I

i s absolutely continuous (with r e s p e c t

to

t h e Lebesgue measure). Thus

and t h e s t o c h a s t i c p r o c e s s IMt,

t

^E R+j h a s t h e s a m e family of finite dimensional distributions as INt,

t

^E R+j. However, f o r all o, t h e realizations

t

k &(o) are everywhere discontinuous, n e i t h e r monotone n o r integer-valued!

W e are basically in t h e same situation as in Example 2.1. The realizations of IN

j

all lie in

Ix E RCO")lx : [0, -) ^-4

N

, x(s) 4 x ( t ) whenever s 4 t ] which does not belong

to

RT.

All of this comes from t h e f a a t t h a t a subset B of RT c a n n o t L i e in RT u n l e s s t h e r e e x i s t s a countabLe subset S fl T w i t h t h e p r o p e r t y : fJ x E B a n d x ( t )

=

y ( t ) for aLL

t

in S t h e n y E B 13, Theorem 36.3

1.

This means t h a t any set of t h e t y p e Ix E XZT(x(t) E F f o r all

t

E T'

c

TI

where F

c

R i s closed, are not necessarily in RT, s i n c e they usually cannot b e ob- tained as countubLe intersections of s e t s in RT. This i s especially important when i t comes

to

t h e study of functionals of stochastic processes. For a s t o c h a s t i c p r o c e s s

IXt,

t

^E TI, let

then f o r a l l a E 8 ,

where

S,

=

Ix E

r(

f o r a l l

t

E T , x ( t ) r a ]

,

but S, g

9,

and t h u s J i s not even measurably r e l a t e d

to

t h e s t o c h a s t i c p r o c e s s IXtj. This point i s brought home by considering t h e two equivalent p r o c e s s e s of Ex- ample 2.1. Here, both

J1 :

=

inft ,TYt, a n d J; : inft, TY;

t u r n o u t

to

b e m e a s u r a b l e functions from

n

i n t o [0, 11 b u t in n o way "equivalent", s i n c e

These are some of t h e simplest examples w e know t h a t c l e a r l y s u g g e s t t h a t t h e class

RT

is o f t e n too small to o b t a i n an a p p r o p r i a t e p r o b a b i l i s t i c d e s c r i p t i o n of s t o c h a s t i c p r o c e s s e s . The a p p l i c a t i o n s should, of c o u r s e , d j c t a t e t h e framework t o u s e in a n y p a r t i c u l a r s i t u a t i o n . In t h e n e x t s e c t i o n s , w e show t h a t there is a r a t h - er g e n e r a l a p p r o a c h t h a t allows

us to

avoid some of t h e o b j e c t i o n s t h a t o n e may h a v e

to

t h i s "simple" definition of s t o c h a s t i c p r o c e s s e s .

3. SOME TOPOLOGICAL CONSIDERATIONS

From a topological viewpoint, t h e shortcomings of t h e "finite dimensional dis- t r i b u t i o n s " d e s c r i p t i o n of s t o c h a s t i c p r o c e s s e s come from t h e f a c t t h a t lZT d o e s n o t t a k e i n t o a c c o u n t t h e underlying topology of T. The o-field

RT

is n o t in g e n e r a l a Borel f i e l d , although t h e f i r s t s t e p in t h e c o n s t r u c t i o n of RT i s topological in na- t u r e . W e c a n think of RT as g e n e r a t e d by t h e c l a s s of m e a s u r a b l e r e c t a n g l e s

as (ti,

. . .

, t k ) r a n g e s o v e r 1 0 ) a n d t h e G i r a n g e s o v e r G(R), t h e o p e n s u b s e t s of R .

This c l a s s of m e a s u r a b l e r e c t a n g l e s is t h e b a s e f o r t h e p r o d u c t topology on lZT b u t in g e n e r a l RT i s n o t t h e Borel field with r e s p e c t

to

t h e p r o d u c t topology.

Unless T is a countable s p a c e , t h e p r o d u c t topology h a s n e v e r a c o u n t a b l e b a s e 16, Theorem 63. If B, d e n o t e s t h e Borel field g e n e r a t e d by t h e open sets of t h e p r o d u c t topology, we h a v e t h a t

with equality i f

T

i s c o u n t a b l e . F o r example, if T c

R

i s a n open i n t e r v a l , l e t A b e t h e s u b s e t of lZT t h a t c o n s i s t s of t h e c o n s t a n t functions with v a l u e s in [0.1]. Then A belongs

to

B , b u t n o t

to

RT.

The "classical" a p p r o a c h essentially ignores t h e topology with which T is en- dowed, in f a v o r of t h e d i s c r e t e topology. And since, with r e s p e c t

to

t h e d i s c r e t e topology, all functions in

R~

are aontinuous, t h e r e i s no way

to

distinguish between those realizations t h a t w e identify as continuous (with r e s p e c t

to

t h e usual topolo- gy o n R ) a n d any o t h e r realizations, t h a t are also "continuous" b u t now with r e s p e c t

to

t h e d i s c r e t e topology.

One g e n e r a l a p p r o a c h , t h a t allows us

to

include ( a t least

to

o u r knowledge) all interesting s t o c h a s t i c p r o c e s s e s , and which s k i r t s a r o u n d all of t h e i n h e r e n t diffi- culties of t h e "classical" a p p r o a c h , i s to think of s t o c h a s t i c p r o c e s s e s as random lower ( o r u p p e r ) semicontinuous functions. The realizations of s u c h p r o c e s s e s are t h e n lower ( o r u p p e r ) semiaontinuous functions, a r a t h e r l a r g e class of functions t h a t should include n e a r l y all possible applications. And f o r t h i s class, t h e r e i s a n a t u r a l choice of topology, a n d a n a p p r o a c h t h a t avoids most of t h e pitfalls of t h e

"finite dimensional distributions" a p p r o a c h .

F o r a n y function x : T

--. a,

^{t h e epigraph of x is} t h e s u b s e t of t h e p r o d u c t s p a c e T X Rdefined by

To any s t o c h a s t i c p r o c e s s fXt,

t

E Tj w e c a n a s s o c i a t e i t s epigraphical represen- tation, i.e., t h e set-valued map defined as follows:

o h e p i X ( o )

=

[ ( t , a ) l a

a

Xt(o)j

.

For a n y finite set I

=

[(ti, a,),

. . . ,

(tq, a q ) l in T X Re we have

Since R T i s t h e minimal u-filed g e n e r a t e d by sets of t h e t y p e

with I ( T X R ) t h e finite s u b s e t s of T X R . From (3.1) i t also follows t h a t

The

sets

of I ( T X

R)

form a b a s e f o r t h e d i s c r e t e topology of T X R , and t h e y are also compact with r e s p e c t

to

t h i s topology.

In t h e epigraphical view, t h e "classical" approach defines a stochastic pro- cess IXL,

t

E TI with domain (Q, A , p ) as a measurable map from (Q, A ) into (ItT, RT), where measurability means t h a t

f o r all subsets K of T X R , t h a t are compact with r e s p e c t

to

t h e d i s c r e t e topology.

This highlights t h e s o u r c e of t h e limitations of t h e classical approach, i t is not able t o identify t h e topological p r o p e r t i e s of t h e realizations beyond those t h a t can be identified by t h e d i s c r e t e topology. The preceding relation also suggest t h e remedy t o use, in o r d e r

to

bring t h e topology of T into t h e probabilistic descrip- tion of t h e process. Instead of working with t h e d i s c r e t e topology on T X R, w e aould equip T X R with a topology t h a t would be more a p p r o p r i a t e f o r t h e applica- tion

at

hand.

Let us r e t u r n t o Example 2.1 with T

=

R,. If P and P' denote t h e probability measures induced by Y and Y' respectively, then

f o r all subsets K of T X

E

t h a t are compact f o r t h e d i s c r e t e topology. The situation i s completely different if compact refers

to

t h e "natural" topology, i.e. t h e usual topology on relative

to

T X

E.

I t i s easy t o verify t h a t f o r any

8

^E (- 1 , 0) and

[al, a2]

c

T, w e have t h a t

and

This time, t h e "induced" probability measures will be different but of course they aannot be defined on RT, t h a t in t h e classical approach i s t h e "universal" function- a l s p a c e f o r dealing with stochastic processes.

4. SEPARABILITY.

MEASURABILITY

AND STOCHASTIC EQUIVALENCE

The epigraphical a p p r o a c h focuses i t s attention on t h e sets of t h e type:

to

define measurability, as w e l l as

to

s e r v e as building block in t h e definition of t h e probability measure associated with t h e p r o c e s s [Xt,

t

^{E Tj. Let}

K,

denote t h e class of compact subsets of T X

R

where T is the product topology generated by

r1

on T and t h e usual topology on

R.

M e a s u r a b i l i t y of t h e p r o c e s s [Xt,

t

^E Tj will now mean:

for

all K in

K,,

This condition i s closely r e l a t e d

to

t h e classical notion t h a t t h e p r o c e s s i s measur- able, which means t h a t

(o, t) k Xt(o) i s A 8 B(T)

-

measurable (4.2)

where B(T) i s t h e Bore1 field on T generated by t h e

rl-

open sets. In Section 6, w e shall show t h a t f o r stochastic processes with lower semicontinuous realizations, these

t w o

conditions are equivalent. W e bring this f a c t to t h e f o r e

at

this time, be- cause

to

r e q u i r e t h a t a p r o c e s s be measurable is a standard condition used

to

overcome some of difficulties c r e a t e d by t h e classical definition. By definition any stochastic process is R~-measurable, but not neaessarily in

terms

of (4.2) or (4.1).

This follows from t h e f a c t a l l sets t h a t are compact with r e s p e c t

to

the d i s c r e t e to- pology are also T- compact.

Closely r e l a t e d

to

t h e notion of measurability of a stochastic p r o c e s s is t h a t of t h e s e p a r a b i l i t y of a stochastic process, as introduced by Doob. Among t h e

ma-

j o r shortcomings of t h e class R~ is t h e f a c t t h a t subsets of t h e type

where F is a closed s u b s e t of

R,

do not necessarily belong

to z.

One circumvents t h e potential difficulties by requiring t h a t t h e stochastic process [Xt,

t

^E Tj b e separable, i.e. t h e r e exists a n everywhere dense countable subset D of T and a p- null

set

N

c

Q such t h a t f o r e v e r y open set G

c

T and closed subset F of

R,

t h e sets

[o E QJXL(o) E F f o r a l l

t

^E G

n

^Dl

and

[ o E QIXt(o) E F f o r all

t

E G ~

d i f f e r from e a c h o t h e r at most on a subset of N [5

1.

In terms of t h e realizations of t h e s t o c h a s t i c p r o c e s s IXt,

t

^E Tj, separability means t h a t f o r all o E Q \ N, t h e function

t

k Xt(o) i s D-separabLe [3

1,

i.e. f o r e v e r y

t

in T t h e r e e x i s t s a sequence It,, n

=

I,,..

.

j such t h a t

tn

E D,

t =

l i m t n , and Xt(o)

=

lim Xtn(o)

,

n n

in o t h e r words, f o r e v e r y o E Q \ N, t h e realization i s completely determined by i t s values on D. A s t o c h a s t i c p r o c e s s s e p a r a b l e with r e s p e c t

to

D i s

R ~ -

measur- able, a n d one may reasonably assume t h a t t h e f a c t t h a t D i s countable removes t h e

"discrepancies" connected with "uncountabilities". Of c o u r s e not all s t o c h a s t i c p r o c e s s e s are s e p a r a b l e . Process [Yt] of Example 2.1 i s not s e p a r a b l e , although t h e equivalent s t o c h a s t i c p r o c e s s IYC] i s s e p a r a b l e . In f a c t , given any finite- valued p r o c e s s t h e r e always e x i s t s a n equivalent p r o c e s s defined on t h e same pro- bability s p a c e t h a t i s s e p a r a b l e [3, Theorem 38.1.

].

A t f i r s t , i t may a p p e a r t h a t i t i s possible

to

r e s t r i c t t h e study of stochastic p r o c e s s e s

to

those t h a t are s e p a r a b l e , b u t t h e r e i s some hidden difficulty. S e p a r a - bility i s defined in term of a r e f e r e n c e set D. F o r t h e convergence of s t o c h a s t i c processes, i t would be n e c e s s a r y

to

p r o v e f i r s t t h a t t h e r e e x i s t s

a

set D with r e s p e c t

to

which all elements of t h e sequence ( o r net), as w e l l as t h e limit process.

are s e p a r a b l e . Moreover, t h e existence of a n equivalent s e p a r a b l e p r o c e s s does not mean t h a t t h e functionals defined on t h e s e p r o c e s s e s will in any way b e com- p a r a b l e ; think a b o u t t h e p r o c e s s e s IYtj and IYi] of Example 2.1 and t h e s u p func- tional, see Section 2. Separability only g u a r a n t e e s t h a t sets of t h e t y p e (4.1) are measurable and t h a t t h e i r probability c a n b e determined by t h e family of finite di- mensional distributions. If IXt,

t

E T] i s not s e p a r a b l e , nothing c a n b e said a p r i o r i about

sets

of t h e type (4.1), and no additional information i s gained from t h e f a c t t h a t t h e r e i s a n equivalent s e p a r a b l e s t o c h a s t i c process. Thus a functional of t h e s t o c h a s t i c p r o c e s s involving sets of t y p e (4.1) cannot b e analyzed in terms of t h e same functional defined on a n equivalent stochastic process.

Roughly speaking, s e p a r a b i l i t y i s a n attempt

at

recovering t h e topological s t r u c t u r e of T,

at

posteriori. The a p p r o a c h developed in t h e n e x t sections t a k e s t h e topological s t r u c t u r e of T d i r e c t l y into account.

5. THE EPIGRAPHICAL APPROACH

The e a r l i e r sections have pointed out t h e shortcomings of t h e "classical" ap- proach by reformulating i t in terms of t h e epigraphical r e p r e s e n t a t i o n of t h e pro- aess. W e have s e e n t h a t t h e i n h e r e n t weaknesses of t h i s a p p r o a c h c a n be overcome by requiring t h a t t h e s t o c h a s t i c p r o c e s s satisfy t h e s t r o n g e r measurability condi- tion

t o € fllepi X ( o )

nK

f

41

€ A f o r a l l

K

EX, _(5.1) which t a k e into account t h e topological s t r u c t u r e of T.

All t h a t follows i s devoted t o t h e study of s t o c h a s t i c p r o c e s s e s t h a t satisfy condition (5.1) and have lower semicontinuous (1.sc.) realizations, i.e.,

t

k Xt(o) is 1-sc. on T, f o r all o ^E fl

.

Such s t o c h a s t i c proaesses, with possibly t h e values

+ -

^and

^-

^{-, are called r a n -}

dom L.sc. j b n c t w n s . In a n o t h e r setting, such functions are known as normal i n - tegrands, and much of t h e t h e o r y developed by Rockafellar [9, 101 f o r normal in- t e g r a n d s c a n be transposed

to

t h e p r e s e n t context. Many of t h e questions r a i s e d in t h e e a r l i e r sections seem t o find t h e i r natural formalization in terms t h e p r o p e r - ties of random l.sc functions and t h e associated epigraphical behavior. This leads

us

a l s o

to

consider t h e associated random closed set

o k e p i X ( o ) : n 3 R

.

(5.3)

For e a c h o , t h e set e p i X ( a ) i s a closed s u b s e t of T X

B

sinae t h e functions

t

k Xt(o) are l.sc., and t h e measurability of this set-valued function follows from condition (5.1).

All of this suggests defining a topology f o r t h e s p a c e of (extended real- valued) l.sc. functions in

terms

of t h e epigraphs, t h e mi-topology. W e s h a l l see t h a t t h e corresponding Bore1 field provides us with t h e desired i n t e r p l a y between topological p r o p e r t i e s and measurability. W e follow t h e development t h a t w a s ini- tiated in [I21 and review h e r e s o m e of t h e main f e a t u r e s of t h a t theory.

At f i r s t i t may a p p e a r t h a t t h e requirement t h a t t h e p r o c e s s h a s l.sc. paths i s a r a t h e r s e r i o u s limitation. A t least if we use this framework f o r t h e study of gen- eral s t o c h a s t i c processes. This i s not t h e c a s e . Of c o u r s e , s t o c h a s t i c p r o c e s s e s with continuous realizations f i t into this class, but a l s o any cdd-l& p r o c e s s (con- tinuous from the r i g h t , limits from t h e l e f t ) admits a t r i v i a l modification t h a t makes

i t a s t o c h a s t i c p r o c e s s with 1-sc. p a t h s . Although, we r e s t r i c t o u r s e l v e s

to

t h e 1.sc. c a s e , i t i s c l e a r t h a t all t h e r e s u l t s h a v e t h e i r c o u n t e r p a r t in t h e u p p e r s e m - icontinuous (u.sc.) c a s e , r e p l a c i n g e v e r y w h e r e e p i g r a p h by h y p o g r a p h .

C r u c i a l

to

t h e ensuing development i s t h e f a c t t h a t f o r s t o c h a s t i c processes t h a t are l.sc. random f u n c t i o n s , we c a n i n t r o d u c e a notion of c o n v e r g e n c e which i s n o t only t h e a p p r o p r i a t e o n e if w e are i n t e r e s t e d in t h e e x t r e m a l p r o p e r t i e s of the p r o c e s s , as w e l l as f o r many r e l a t e d functionals, b u t also p r o v i d e s in many s i t u a - t i o n s a m o r e s a t i s f a c t o r y a p p r o a c h to t h e c o n v e r g e n c e of s t o c h a s t i c processes as t h e s t a n d a r d functional a p p r o a c h .

6 .

TFE EPIGRAPHICAL

RANDOM SET

H e n c e f o r t h , w e work in t h e following s e t t i n g

-

^{(fl, A , p) a} c o m p l e t e p r o b a b i l i t y s p a c e ,

- ^(T,

r l ) a locally c o m p a c t s e p a r a b l e m e t r i c s p a c e ,

-

^{(w, t ) k Xt(w) :}

n

x T --*

k

a r a n d o m L.sc. m n c t i o n . By t h i s we mean t h a t

(i) f o r e v e r y o, t h e r e a l i z a t i o n t H Xt(o) i s l.sc. with v a l u e s in t h e e x t e n - ded reak

(ii) t h e map (w, t ) i, Xt(w) is A @B1- m e a s u r a b l e , w h e r e B 1 is t h e Bore1 field o n T.

The a s s o c i a t e d e p i g r a p h i c a l random s e t , i s t h e map

t h a t t a k e s v a l u e s in t h e c l o s e d s u b s e t s of T x

R ,

including t h e empty set.

The p r o d u c t s p a c e T X

R

is given t h e p r o d u c t topology of

r1

with t h e n a t u r a l topology on

R ,

we d e n o t e i t by 7 . Thus (T x R , 7 ) i s a locally c o m p a c t s e p a r a b l e m e t r i c s p a c e . L e t

- F = F(T

X R ) d e n o t e t h e c l o s e d s u b s e t s of T X

R,

-

G

=

G (T X

R )

d e n o t e t h e o p e n s u b s e t s of T X

R ,

-

_K

₌ K(T

x

E)

d e n o t e t h e c o m p a c t s u b s e t s of T X

R ,

For any s u b s e t C of T X

B,

let

The topology T g e n e r a t e d by t h e subbase of o p e n sets jFK, K E

K

j

,

a n d IFc, G E G j

makes t h e topological ( h y p e r ) s p a c e ( F , T) r e g u l a r and compact, see e.g. [ 4 , Propo- sition 3.21. If T h a s a countable base, so does ( F , T), see e.g. [7, Theorem 1-2-11 and [ 4 ] in which case a b a s e f o r T i s given by t h e open sets of t h e t y p e

where c l C denoted t h e c l o s u r e of C, and t h e

oome from a countable b a s e of open

sets

f o r T X

R.

The BoreLfieLd, g e n e r a t e d by t h e T-open s u b s e t s of F , will b e denoted by B ( F ) . I t i s e a s y to see t h a t i t c a n b e g e n e r a t e d from t h e subbase of open

sets

(6.1), and in t h e countable-base case by t h e r e s t r i c t e d class (6.2), cf.,

n, 111.

W e c a n also view t h e e p i g r a p h i c a l random set as a random v a r i a b l e defined o n fl and values in E , t h e s u b s e t of F , c o n s i s t i n g of

the sets that

are epigraphs. I t i s e a s y

to

v e r i f y t h a t E i s a closed s u b s e t of F , and t h u s with t h e T-relative topolo- gy, i t i n h e r i t s a l l t h e p r o p e r t i e s of F. The map

o h epi X ( o ) : f l + E

i s m e a s u r a b l e (is a r a n d o m s e t ) , if f o r all K E

K.

(epi X )-'(K)

= lo

^{E CZ(epi} X ( o )

n

^{K # # j E A}

.

This i s equlvalent [ 9 ,

111 ^to

a n y one of t h e following conditions:

(epi X

)-'(F)

E A f o r a l l F E F ,

(epi X )-'(B) E A f o r all closed balls B of T x

R,

o k epi

X.

( o ) admits a Castaing r e p r e s e n t a t i o n (see below),

g r a p h (epi X ) E A 8 B 1 ,

o k e p i X ( o ) :

L?

-, F i s B(F)-measurable.

E a c h o n e of t h e s e c h a r a c t e r i z a t i o n s c a t c h e s a s p e c i a l a s p e c t of t h e measurability of t h e e p i X

.

To h a v e measurable g r a p h c o r r e s p o n d s to having [ X t ,

t

^E T j a m e a s u r a b l e s t o c h a s t i c p r o c e s s . The f a c t t h a t t h e r a n d o m (closed) sets admits a Castaing r e p r e s e n t a t i o n g e n e r a l i z e s t h e notion of s e p a r a b i l i t y of a s t o c h a s t i c pro- cess. And t h e Last o n e induces on ( F , B ( F ) ) , more p r e c i s e l y on ( E , B ( E ) ) , a d i s t r i - bution. From t h e definitions, i t i s immediate t o v e r i f y

[lo,

P r o p o s i t i o n

11

t h a t

THEOREM 6 . 1 The s t o c h a s t i c p r o c e s s [ X t ,

t

^E T J w i t h L.sc. r e a l i z a t i o n s i s m e a s u r a b l e i f a n d o n l y i f ( o , t ) k Xt(o) i s a r a n d o m 1.sc. m n c t i o n , o r s t i l l , i f a n d o n l y i f o k e p i X ( o ) i s a r a n d o m closed s e t .

A c o u n t a b l e c o l l e c t i o n of measurable functions lxk. a k ) :

n

^-, T X

B,

k

=

1 ,...

1

i s a C a s t a i n g r e p r e s e n t a t i o n [9] of epi X E A if

and f o r a l l o E e p i X ,

We now show t h a t t h e f a c t t h a t t h e random closed set e p i X admits a Castaing r e p r e s e n t a t i o n is a n e x t e n s i o n of t h e notion of s e p a r a b i l i t y f o r t h e s t o c h a s t i c pro- cess l X t ,

t

^E TI. The k e y f a c t is t h e following:

THEOREM 6.2 A n y r e a l - v a l u e d s e p a r a b l e s t o c h a s t i c p r o c e s s I X t ,

T

E

Tj

w i t h 1.sc.

r e a l i z a t i o n s i s a m e a s u r a b l e p r o c e s s .

PROOF In view of Theorem 6.1, and t h e equivalent definitions of measurability ( f o r a random s e t ) , i t s u f f i c e s to e x h i b i t a countable c o l l e c t i o n of m e a s u r a b l e functions

s u c h t h a t f o r a l l o

r n,

Suppose D

=

Idi, i E I

j c

T i s t h e countable s e t with r e s p e c t t o which iXt,

t

E TI is separable, and let A

=

la,, j E J

j

be a countable dense subset of

R.

Then D X A i s a countable dense subset of T X

R.

Let l(xl. a,): R -+ T X

R,

i E I, j ^E J ] be a count- able collection of random functions defined by

Since lXt, t E T

j

i s a stochastic process, f o r all (i, j)

Let N be a p-null subset of Q such t h a t e v e r y realization of X ( o ) is D-separable f o r a l l o E R \ N. W e have t h a t f o r all o E R \ N,

epi X.(o) = c l ( e p i X.(o)

n

l(xi(o>, a,(o)), i € 1 , j E J j

.

For all o E R \ N and a l l

(x,

a ) E epi X ( a ) , by D-separability of [Xt,

t

^E Tj, t h e r e exists Id, E D, n

=

1 ,

... j

such t h a t

x

=

limndn, and a

r

Xt(o)

=

limnXh(o)

.

Since A i s dense in R a n d

t

k Xt(o) i s l.sc.,

w e

c a n always find a sequence

la,

^{E A,}

n

=

1 ,

... ^j

such t h a t a,

r

Xh(o) and a

=

limn a,. This means t h a t f o r a l l o E C2 \ N

But this yields equality since t h e r e v e r s e direction is trivially satisfied.

The stochastic p r o c e s s l X i ,

t

^E Tj having epigraphical representation

is measurable, by Theorem 6.1, i.e., f o r all K E

K

lo

^E R l e p i X 1 ( o ) n K $ 0 1 E A

.

For t h e process [Xt,

t

^E T 1,

w e

have t h a t

a l s o belongs t o A . The latter set i s of measure 0 and belongs

to

A , since ( n , A , p ) i s complete by assumption.n

The c o n v e r s e of t h i s theorem does not hold. A counterexample would b e t h e p r o o e s s IYt,

t

^E T

j

as defined in Example 2.1 with T

= R,.

REMARK 6.3 In Section 4, w e indicated t h a t s e p a r a b i l i t y w a s introduced

to

recov- e r t h e measurability of t h e sets

Io E n ( X t ( o ) E F, f o r all

t

E G

c

T j

where F

c R

i s closed and G i s s p p e n , w e should note t h a t t h e r e are of c o u r s e no measurability problems if ( a , t ) b Xt(o) i s a random l.sc. function. And thus in t h a t context, separability i s mostly a n i r r e l e v a n t concept.

7. DISTRIBUTIONS AND DISTRLBUTION FUNCTIONS

In section 6 , w e have s e e n t h a t

to

e a c h random l.sc. function w e c a n associate a n epigraphical random closed

set.

A s w e shall show now,

to

e a c h random closed

set

t h e r e c o r r e s p o n d s a distribution function, which in t u r n will allow us

to

define t h e

"distribution function" of a random l.sc. function. Let us denote by

r

^{a random}

closed s e t , defined on

n

and with values in t h e closed s u b s e t s of T X R . Let P denote t h e distribution of r , i.e., t h e probability measure induced on B ( F ) by t h e relation

f o r a l l B ^E B ( F ) .

Since t h e topologial s p a c e (F, B ( F ) ) i s metrizable, see Section 6 , e v e r y proba- bility measure defined on B ( F ) i s r e g u l a r [2, Theorem 1.11, and thus i s completely determined by i t s values on t h e open ( o r closed) s u b s e t s of F. If w e assume t h a t F h a s a countable b a s e

-

and f o r t h i s i t suffices t h a t T h a s a countable base

-

e v e r y open

set

in F i s t h e countable union of elements in t h e base, obtained by taking fin-

i t e intersection of t h e elements in t h e subbase. Thus, i t will c e r t a i n l y b e sufficient

to

know t h e values of P on t h e subbase (6.1) t o completely determine P. This obser- vation will bring us

to

t h e notion of a distribution function f o r t h e random closed

set r

^[12].

F i r s t o b s e r v e t h a t t h e r e s t r i c t i o n of

P to

t h e c l a s s IFK, K E K j defines a func- tion D on K through t h e relation:

f o r all K E K. This function h a s the following properties:

f o r any decreasing sequence tK _,, u

=

^1,.

. .

^{j in}

^{K ,}

t h e sequence

(7.4) ( K)

v =

^{1 . .} j decreases t o D(1im K _), ;

f o r any sequence of sets [K,

v =

0

,...

j, the functions

[A,,

n

=

0, 1

,...

^{j defined} re- cursively by

and f o r n

=

2,

...

take on t h e i r values [0, 11.

The p r o p e r t i e s of D on K are essentially t h e s a m e as those of t h e distribution function of a 1- _or n-dimensional random variable. P r o p e r t y (7.4) i s t h e same as right-continuity, whereas (7.3) corresponds to t h e continuity at

-

^{0 f o r a distribu-}

tion function on t h e r e a l line. P r o p e r t y (7.5) can be viewed as a n extension of t h e notion of monotonicity. In view of this, and the fact 112, Choquet's Theorem 1.31 t h a t any function D:K

--.

[0, 11 t h a t satisfies t h e conditions (7.3). (7.4). (7.5) uniquely determines a probability measure on

B(F),

w e call D t h e d i s t r i b u t i o n f i h c t i o n of l?.

The f a c t t h a t w e can r e s t r i c t t h e domain of definition of D t o t h e subclass

P b

of K is v e r y useful in a number of applications, where KUb

=

Ifinite union of closed balls with positive radii j;

note t h a t $ E as t h e union of an empty collection. This comes from t h e f a c t t h a t t h e properties of

(F,

T) enables us t o generate

B(F)

from t h e family

in fact, f o r all K E K ,

w e

have

and

and consequently

D(K)

=

P(FK)

=

inf K , , P(FK,)

=

inf K, ,KD(K')

.

K C

A

^KClVub

The (probability) d i s t r i b u t i o n fLLnction of a random lower semicontinuous function ( 0 , t ) k Xt(o) i s t h e distribution function of i t s epigraphlcal random set.

Since t h e random

set

t a k e s i t s values in t h e (hyper)space of e p i g r a p h s w e could reformulate i t in t h e following

terms: let

C be a ri-compact s u b s e t of T, and a E

R ,

then

D(C, a ) :

=

p [ o ( i n f t E c X t ( o ) 4 aj

defined on (the compact s u b s e t s of T) X

B

c a n b e used instead of t h e usual defini- tion of D on t h e compact s u b s e t s of T x

B.

8.

. . . ^AND

^IWUTJZ

DIBENSIONAL DISTRIBUTIONS!

Let us consider [Xt,

t

E Tj a measurable s h h a s t i c p r o c e s s with l.sc. realiza- tions, then e p i X : fi

3

T X

B

i s a closed random set with distribution function D : K --, LO, 11. Any finite set I

=

[(ti, a l )

, . . . ,

(th, a h )

j c

T X

B

i s r-compact, and thus w e have

In p a r t i c u l a r , if w e fix t , then f o r all a ^E R

where P t r e f e r s to t h e l-dimensional probability measure of t h e random variable Xt. Similarly, if w e fix tl,

. . .

^{, tq, then}

I t i s now immediate t h a t

THEOREM 8.1

U

[Xt, t E

T

j i s

a measurable stochastic process w i t h L.sc. realiza- t i o n s , t h e f i n i t e d i m e n s i o n a l d i s t r i b u t i o n s a r e completely determined b y

D,

or e q u i v a l e n t l y b y t h e r e s t r i c t i o n o f D t o the f i n i t e s u b s e t s o f T

X

R.

Of c o u r s e , t h e c o n v e r s e of t h i s theorem does not necessarily hold. Take f o r exam- ple t h e p r o c e s s [Yt,

t

E

TI

of Example 2.1 with T

= R+

and let

1 3

K

=

[t,,

t,]

x [-

- -

^{where 0}

< tl < t2.

Then D(K)

=

plulV(u) E

[t,. t2]j >

0, 2 '

but D(1)

=

⁰ f o r a n y finite s u b s e t I of K. The family of finite dimensional distribu- tions, t h a t assigns a value

to

D f o r e v e r y finite s u b s e t of K, does not a l l o w

us to

make a n y inference a b o u t t h e value to assign

to

DO().

REMARK 8.2 N o t e t h a t t h e s t a n d a r d consistency conditions f o r t h e family of finite dimensional distribution could actually b e derived from t h e "monotonicity" p r o p e r - ty (7.5) of t h e distribution function D. Thus, we c a n think of t h i s family of finite di- mensional distributions itself as a distribution function, but defined on t h e finite s u b s e t s of T X

R.

This suggests a n o t h e r a p p r o a c h

to

Kolmogorov's Consistency Theorem via Choquet 's Theorem.

The f a c t t h a t a compact

set

K

c

_T X

R

cannot b e obtained as a countable union of finite

sets

i s a topological f a c t t h a t leads

to

a probabilistic discrepancy in t h e example involving t h e p r o c e s s [Yt,

t

^E

T j.

DEFINITION 8.3 The

d i s t r i b u t i o n j b n c t i o n of a r a n d o m Z.sc. j b n c t i o n

^is

said t o be

inner s e p a r a b l e , f f

t o a n y

K € K

and

E

>

^0,

there corresponds a f i n i t e set

I,

c

K

s u c h t h a t

D(K)

<

^DO,)

+

^{E .}

The basic difference between separability of a s t o c h a s t i c p r m e s s and t h e inner s e p a r a b i l i t y of i t s distribution i s t h a t s e p a r a b i l i t y is aimed

at

t h e reconstruction of

sets

through "finite sets", whereas i n n e r s e p a r a b i l i t y is aimed

at

t h e recon- s t r u c t i o n of t h e probabilistic content of t h e

sets

in terms of the probability associ- a t e d

to

finite sets.

PROPOSITION 8.4 [12, Proposition 4.61.

Shppose

[Xt,

t

^E Tj

i s

a

measurable sto-

c h a s t i c process w i t h Z.sc. realizations. U i t

is

separable, then its d i s t r 3 b u t i o n

j b n c t i o n

i s

i n n e r separable. Moreover,

f f

its d i s t r i b u t i o n f i n c t i o n

i s

i n n e r

separable, i t s v d u e s o n

K

a r e completely d e t e r m i d b y its v a l u e s o n the f i n i t e

s u b s e t s o f T

x

R.

This

last

a s s e r t i o n i s an immediate consequence of t h e definition of inner s e p a r a - bility.

9.

WEAK CONVERGENCE AND CONVERGENCE IN DISTRIBUTION

W e show t h a t f o r random l.sc. functions, weak convergence of t h e probability measures c o r r e s p o n d s

to

t h e convergence of t h e distribution functions

at

t h e

"continuity"

sets.

By v, w e index t h e members of a sequence of s t o c h a s t i c processes, t h e in- duced probability measures on B ( E ) , o r t h e corresponding distribution functions on

K = K(T

X

R);

by B ( E ) w e mean t h e Bore1 f i e l d B ( F ) r e s t r i c t e d t o E. With v

=

^w,

o r simply without index, w e r e f e r

to

t h e limit element of t h e sequence. W e have s e e n t h a t f o r e v e r y K E

K:

Since EK i s a closed s u b s e t of F--E i s a closed s u b s e t of F - - , w e c a n easily obtain from t h e Portemanteau Theorem [2] t h a t

PROPOSITION 9.1 U P v converges weakLy to P, t h e n j o r aLL K E K lim supDV(K)

s

D(K)

.

v--+-

Unless P(bdy E K )

=

0, t h e probability measure a t t a c h e d

to

t h e boundary of EKs w e cannot g u a r a n t e e t h a t

lim inf Dv(K) _v--+- 2 D(K)

.

i.e. unless K i s

a

"continuity" point of D in a

sense to

b e defined below. Note t h a t

"continuity

sets"

of D must c o r r e s p o n d

to

P-continuity

sets

and t h a t t h e class of sets f o r which t h i s continuity i s defined must

at

least b e a convergence determin- ing class [Z].

DEFINITION 9.2 An increasing sequence [K", n

=

I,..

.

^j o j compact s e t s is said to repulardy converge to K

ir

K

=

cl

U,"=~K"

a n d int K C u L 1 K n ; (9.3) where i n t S denotes the interior fl the set S .

DEFINITION 9.3 A

d i s t r i b u t i o n fiLnction

D : K -+ [0,1]

i s

distribution-continuous

at

K,

U f o r e v e r y r e g u l a r l y i n c r e a s i n g sequence

[Kn, n

=

1 ,

... j t o

Kt D(K)

=

lim D(Kn)

n ^+-

The

d i s t r i b u t i o n - c o n t i n u i t y set

C D of D, is t h e s u b s e t of K o n which D is distribution-continuous

PROPOSITION 9.4 112, Lemma

1.111. For

a n y K E K , (1)

V

(P(bdy E K )

=

0 ,

then

K E CD;

(if) K E CD

a n d

K

=

c l ( i n t K),

then

P(bdy E K )

=

0.

Assuming t h a t (T, r l ) h a s a c o u n t a b l e b a s e , l e t

~ ; i ~

^C

Pb

b e s u c h t h a t K$' is t h e f i n i t e union of balls t h a t d e t e r m i n e a c o u n t a b l e b a s i s f o r

(T

X B, r ) . W e h a v e

a n d if T h a s a c o u n t a b l e b a s e

This allows us

to

r e p h r a s e weak-convergence of p r o b a b i l i t y m e a s u r e s in

terms

of t h e pointwise c o n v e r g e n c e of t h e d i s t r i b u t i o n functions.

THEOREM 9.5 [12, Theorem

1.151 For t h e jtcmily of random l.sc. f u n c t i o n s

tX

,

X ', v

=

1 ,

...

^j,

e q u i v a l e n t l y of measurable stochastic processes w i t h l.sc. reali- zations, we h a v e that t h e

P'

converge w e a k l y to

P

V and o n l y Ufbr

all

K

E

Pb

n

^CDt

(and U

^{(T, r1)}

has a countable base,

fbr all K E

~l~ n

^CD):

D m )

=

lim DY(K)

.

w-b-

W e r e f e r

to

t h i s t y p e of c o n v e r g e n c e , as

convergence in d i s t r i b u t i o n

of t h e

sto-

c h a s t i c p r o c e s s e s IX:,

t

^t TI

to

[Xt,

t

^E TI, a n d d e n o t e i t b y XeY

Ad

X..

10. CONVERGENCE

IN

DISTRIBUTION

AND

CONVERGENCE OF THE FINITE DIMENSIONAL DISTEUBUTIONS

In t h e classical a p p r o a c h to t h e study of s t o c h a s t i c p r o c e s s e s , convergence of stochastic p r o c e s s e s i s defined in terms of t h e convergence of the finite dimen- sional distributions, t h a t w e denote by

In view of t h e comments in Section 8 , w e cannot e x p e c t t h a t X

sd

^X implies t h a t

1 d

X v 4 X , but t h e c o n v e r s e could reasonably b e conjectured, see Theorem 8.1.

However, in g e n e r a l also t h i s implication fails. The r e a s o n i s t h a t f o r finite sets K

c

K, t h e notions of distribution-continuity and continuity of t h e corresponding finite dimensional distribution d o not coincide.

REMARK 1 0 . 1 This c a n all b e t r a c e d back

to

t h e relationship between t h e epi- topology and t h e pointwise-topology. Equivalence i s obtained in t h e p r e s e n c e of equi-semicontinuity [12, Section 3

1,

see also [4] f o r details.

The passage from convergence in distribution

to

convergence of t h e finite di- mensional distributions and vice-versa, i s based on t h e possibility of "approximat- ing" t h e values of t h e distribution function f o r compact sets K by finite s e t s , in- d e p e n d e n t of v, a n d conversely.

DEFINITION 10.2 The f a m i l y of d i s t r i b u t i o n f i n c t i o n s ID; Dv

=

1 ,

... 1

o n K i s equi-outer r e g u l a r a t t h e f i n i t e set I

c

T X

R,

to e v e r y E

>

⁰ t h e r e c o r r e s p o h d s a compact set K, ^E K " ~

n

^C,, w i t h Kc 3 I s u c h t h a t for v

=

1 ,

...

Dv(K,) < D v ( I )

+

^{E ,} and DO(,) < D O )

+

^E

.

Now, let Ciad. denote t h e finite subsets of T x

F&

i.e.

Cl.d C 11

=

l(tll a l l ,

- . .

^, (tq. a q )

1.

^{q finite}

1

^,

such t h a t t h e distribution function of t h e v e c t o r (XL,,

. . .

, XQ i s continuous

at

( a l ,

. . . ,

a q ) .

DEFINITION 10.3 The f a m i l y of d i s t r i b u t i o n f i n c t i o n s ID; D ", v

=

1,.

. .

^j o n

K

_{i s}

equi-inner separable a t K E

K,

i f to e v e r y o

>

0, t h e r e c o r r e s p o n d s a f i n i t e set I, s u c h t h a t

D(K)

<

^D(1,)

+

o, and DV(K)

<

^DV(I,)

+

^o

for v

=

1,.

. .;

see & f i n i t i o n 8.3.

THEOREM [12, Corollary 4.61 SLLppose [X ;

x

^{", v}

=

1,.

. .

j is a collection of r a n d o m

1.d. r d

L.sc. j b n c t i o n s . Then X

"

^-+ X i m p l i e s

x "

⁴ X a n d o n l y i f [D, D ", v

=

1,.

. . 1

I d i.d.

is equi-outer reguLar o n Cred.. And X " ⁴ X impLiSs X." ^--, X i f and onLy i f ID; D ", v

=

1 ,

...

j i s e q u i - i n n e r separabte.

11. BOUNDED RANDOM LSC. FUNCTIONS

Applications usually r e q u i r e s

us to

r e s t r i c t o u r attention

to

a subclass of processes t h a t possess f u r t h e r properties beside lower ( o r upper) semicontinuity.

From the point of view of t h e eqigraphs, this means t h a t , t h e realizations now be- long

to

E' a subset of E. Let T' be t h e relative T-topology on E'. Then t h e topologi- cal s p a c e (E', T') inherits a number of t h e p r o p e r t i e s of (F, T) [6]. In particular, if (F, T) is metric with countable base, then (E', T') i s metric with countable base.

Thus, in principle all t h e earlier

r e s u l t s

still apply t o (E', T'), and t h e theory of weak-convergence on s e p a r a b l e metric spaces can be used t o obtain convergence c r i t e r i a . In particular, recall that:

THEOREM 11.1 Prohorov. The sequence [ P ", v

=

1 ,

... 1

of p r o b a b i l i t y m e a s u r e s on B(E') is t i g h t i f and onLy i f e v e r y subsequence c o n t a i n s a f u r t h e r subse- quence

that

w e a k l y converges to a p r o b a b i l i t y m e a s u r e .

This means t h a t t h e sequence IP

",

v

=

1 ,

...

^j i s relatively compact. A subset S of E' i s T'-compact if and only if i t is a T-closed subset of E, see Section 6.

W e now deal with bounded processes. W e use this class

to

illustrate t h e poten- tial application of t h e "epigraphical" approach

to

specific classes of stochastic processes. To begin with, let us observe:

LEMMA 11.2 For a l l a E

R+

E,

=

[epi x ( s u p L E T ( x ( t ) ( 5 a1 c E

i s T-compact. And hence, a n y collection of p r o b a b i l i t y m e a s u r e s P o n B(E') s u c h t h a t for e v e r y E

>

0, t h e r e e z i s t s a ² 0 s u c h t h a t for a l l P' E P

is t i g h t .

PROOF. The f i r s t a s s e r t i o n follows from [4, Section 41 and t h e second one from t h e definition of tightness [8]n

Let

b e t h e s p a c e of e p i g r a p h s associated t o l.sc. functions t h a t are bounded below and above by a+. From Lemma 11.2, and Theorem 11.1, i t follows d i r e c t l y t h a t

PROPOSITION 11.3 A n y c o l l ~ c t i o n P of p r o b a b i l i t y m e a s u r e s o n B(E+) i s t i g h t , a n d hence e v e r y subsequence has a convergent subsequence.

12. AN APPLICATION TO GOODNESS-OF-FIT STATISTICS

Let us consider t h e basic case of independent observations ( t i , C2,.

. . ,

C,,) from t h e uniform distribution on [O, I]. Let

us

define t h e empirical p r o c e s s

t )

-

t , if 0

< t <

1 , otherwise

.

where f o r e v e r y o , ~ " ( o , -) i s t h e empiral distribution (taken left-continuous) determined by t h e sample ( t i ,

. . .

, 4,). The realizations ueV(o) are l.sc. on [0,

11

(with r e s p e c t

to

t h e n a t u r a l topology on R); this comes from t h e f a c t t h a t FY is a left-continuous piecewise constant distribution function on R. I t i s a l s o e a s y t o verify t h a t t h e function

( a , t)k UY(o):[O, I]" X [0,

11

^{-, [- 1 ,}

11

i s measurable. Redefining t h e underlying sample s p a c e t o be [0,1Im, and making t h e obvious identifications, w e have t h a t f o r a l l u

=

1 , .

. .

( a , t ) k Uc(o)

=

[O,

11-

^{X [0,}

11

^{4 [- 1 ,}

11

i s a random l.sc. function. W e are h e r e in t h e c a s e when f o r all v

=

1 ,

...

Moreover, f o r all v, t h e corresponding distribution functions IDV, v

=

1.

...

j are inner s e p a r a b l e

at

K, f o r all K in KUb. This follows from t h e inner-separability of t h e distribution function associated

to

t h e s t o c h a s t i c p r o c e s s IFV(., t ) ,

t

E [O.

111.

Since, w e may as w e l l t a k e f o r balls t h e products of intervals, w e see t h a t epi Fv(o)

n

^{([ti, tZ1 X} [al, a2]) only if FV(o, t2) 6 a l , since FV i s monotone nonde- creasing. Thus f o r any finite collection of balls, t h e value of t h e associated distri- bution function is determined by i t s values on some finite

set.

By Proposition 8.4, and t h e f a c t t h a t t h e values of DV on KUb determine unique- ly i t s values on K , w e know t h a t t h e finite dimensional distributions completely determine DV. Moreover from Proposition 11.3, s i n c e t h e [U;,

t

E T j are (equi-) bounded, t h e associated probability m e a s u r e s are tight. This means t h a t t h e r e al- ways e x i s t s a subsequence

"k

ID , k = l , . . . j converging D ,

Observe t h a t independence did not play any r o l e up

to

now. If t h e

[tk,

k

=

1

,... ^j

are i.i.d, by t h e h w of h r g e numbers, f o r e v e r y I

=

(ti,

. . . ,

tq), t h e finite dimensional distributions converge in distribution t o t h e q-dimensional dis- tribution of t h e random v e c t o r identically zero. And thus t h e limit p r o c e s s

IUt,

t

^E T j

must

b e a s t o c h a s t i c p r o c e s s whose realizations are such t h a t U t ( o ) = O f o r a l l t E [ 0 , 1 ] ,

and f o r all o E Cl \N where N i s a set of measure 0.

Actually a somewhat s t r o n g e r r e s u l t does hold. From, t h e s t r o n g l a w of l a r g e numbers, f o r e v e r y

t

E T

i.e. t h e r e e x i s t s a set N t of measure 0, such t h a t