COEdPACTNESS THEOREMS ON A REAL PROBABILITY SPACE

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COEdPACTNESS THEOREMS ON A REAL PROBABILITY SPACE

By

T.

Matsunawa

S W R Y

A

generalization of Helly's selection theorem to multi-dimensional case

L61

states that any sequence of k-dimensional real random variables {Xn3 (n=1,2,.

. .

⁾

contains a subsequence {Xn}({m}

2

{n}) which converges in the wide sense.

In this article, via expository measure theoretic approach, it is shownthat under certain conditions there exists a subsequence which converges caRpZeteZy to some real random variable in the sense of certain types of convergence.

1.

INTRODUCTION AND PRELIMINARIES

Let (0,F.P) be a basic probability space, and ( R , B , ~ ) designate aninduced probability space by a k-dimensional real random variable X

,

where R being the k-dimensional Euclidean space and B the usual Bore1 field of subsets of R.

Throughout this article k is assumed to be

fitced

independently of n. Further, let F(R,B) be the family of all probability distributions pver the measurable space (R,B)

,

and P(R,B,p) be its subfamily composed by the all absolutely continuous probability distributions with respect to the usual Euclid-Lebesgue measure over (R,B). In the subsequent discussions we shall denote the members of each family by the random variables X, Y,

...

instead of their corresponding probability measures

9,

,'P

. . . ^,

respectively.

Let {Xn} (n=1,2,.

.

^.)and {YnJ (n=1,2,

.

^.)be two sequences of k-dimensional random variables belonging to F(R,B), and

C

be

any

given non-empty subclass of B.

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299 COMPACTNESS THEOREMS

-83

- According t o Ikeda [2] l e t u s consider t h e following types of t h e asymptotic equivalence o r those of convergence f o r r e a l p r o b a b i l i t y d i s t r i b u t i o n s ;

( l o ) any two sequences {Xn} and {Ynl a r e sa>d t o be a~ympt0tiCaZly equi- v a l e n t i n the sense 0.f type ((C))d and denoted by X, ^QYn ((C))d (72 +-I, if f o r every subset E belonging t o C, it holds t h a t

(2O) any two sequences {X,} and {Yn) a r e s a i d t o be asymptoticallyequz- vaZent zn t h e sense o f type (Cld and denoted by Xn ^2.Yn (CJd ( n +

-1.

i f i t holds t h a t

( 1 2 ) _sup

I I P

'n ^(E)

^..

^P

*

^"(E)] ^:^{E c}^{C I}⁺^0, ⁽ⁿ^-;^{a ) .}

Especially, when Y n t s a r e a l l i d e n t i c a l with some f i x e d d i s t r i b u t i o n Y belonging t o F ( R , B ) , we have t h e corresponding notions of t h e convergence t o t h e d e f i n i t i o n s above;

( 1 ) t h e sequence CXnl i s s a i d t o converge i n the sense of type ((C))d and denoted by Xn + Y ((C)ld ( n *-) , i f it holds t h a t Xn ^QYn

((c

)Id

,

and

( 2 ) t h e sequence {Xn) i s s a i d t o converge i n the sense of type (CId and denoted by Xn -t Y (C)d (n+m). i f it holds t h a t Xn Q Yn(c

.

To describe t h e c l a s s i c a l compactness theorem i n terms of t h e p r o b a b i l i t y measures, we introduce another weaker convergence than t h e above ones, w h i c h i s t h e generalized one of convergence i n t h e wzdesensedefined i n Tucker's book 161.

L e t 'Q be a measure over (R,B) induced by some fixed measurable function defined on R t o R, and suppose t h a t 0 <_Q'(E)

2

1 f o r a l l E 6 C We t h e n d e f i n e t h a t

( 3 ) t h e sequence {Xn) i s s a i d t o converge i n the sense of type [[C]Jd and denoted by Xn + Z [[C]ld ( n +

-

⁾

^,

i f for every subset E belonging t o C

,

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X

(1.3) IP

n ( ~ )

-

Q'(E)I ⁺O,

( n

+

-1.

It should be remarked t h a t i s not n e c e s s a r i l y a p r o b a b i l i t y measure over ( R , B ) and hence Z need not be a member of F(R,B). The following r e s u l t shows t h i s unfortunate d i f f i c u l t i e s ; Let

{An> (n=1,2,. . .

⁾ be a monotone non- decreasing sequence of s u b s e t s such t h a t

An

E C f o r each n snd

An +

^{R a s}

n + m . Consider t h e p r o b a b i l i t y d i s t r i b u t i o n defined over (R,C) given by

' n

0 , f o r E C A ~ , P (E) =

1, f o r E C R - A n ,

z ' n

for each

n

Then Q = limn+-&' = O ( t h e zero measure) f o r a l l E ^EC, and hence { X n ) converges t o a c e r t a i n measurable function Z defined R t o R.

This example i l l u s t r a t e s t h e e x i s t e n c e of escaping t h e p r o b a b i l i t y mass t o i n f i n i t y , although it i s an extreme and not a p r a c t i c a l case.

Here, l e t us consider some familiar c l a s s e s a s t h e b a s i c c l a s s C. Let

M

be t h e c l s s of a l l k-dimensional i n f i n i t e i n t e r v a l s which a r e right-opened, S t h e c l a s s of a l l k-dimensional l e f t - c l o s e d and right-opened i n t e r v a l s , and

A

be t h e c l a s s of a l l f i n i t e d i s j o i n t unions of t h e members of S.

Under t h e s e setup we review t h e h i s t o r i c a l r e s u l t s t o make ours c l e a r i n t h e subsequent s e c t i o n s .

C

THEOREM 1.1. ( H e l l y ' s S e l e c t i o n Theorem).

Let

{Xn} (n=1,2,.

. .

⁾

be

any sequence o f k-dzmensionat random varzabZes belonging to

F(R,B).

Then there

e x i s t a subsequence { m ) E { n ) and a measure

Q'

over

(R,B)

induced by some

,fixed measurabZe function

Z

defined on

0

t o

R,

satisfying the condition

0

5

Q'(E) 5 1

,for a22

E E M,

such that

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301

COMPACTNESS THEOREMS

-

⁸⁵

-

This theorem assures us t h a t t h s family F(R,B) is

retativezy compact

i n t h e sense of type

[[Mild ,

and i s a l s o c a l l e d a

oeak compactness theorem.

Under a c e r t a i n condition K S Rao and D . G. Kendall [5] extended t h e theorem t o a

00nrp7,ete compactness

case, which c a n b e r e c a s t i n our fashion a s follows;

THEOREM 1.2.

Let {Xnl (n=l,

2,.

. .

⁾

be any sequence o f random variabZes belonging t o

F(R,B)

, and

" 2 , n

be the second moment o f {Xnl satisfyzng the foltowzng condztion

where K i s a positive constant and

N

i s an integer. Then there exzst a subsequence {m} C

1%)

and a

probability measure

pY over

(R,B)

znduced b y some fired

r e a l random variable

Y, such that

(1 6 ) Xm -+

Y

((MI),

( n - t - )

Under a c e r t a i n condition more general r e s u l t holds. I n order t o t h i s we need a c e r t a i n kind of uniform stochastic boudedness property of thesequence

of random variables. The sequence I X n > (n=1,2,.

. .

⁾ i s s a i d t o have

property

B ( S ) , provided t h a t for any given E > 0, t h e r e e x i s t a bounded subset B = B(E) belonging t o t h e c l a s s S and a p o s i t i v e integer N = N ( & , B ) such tha%

he

terminology 'property B(B)

'

was introduced by [ 2 ] , and some authors c a l l t h e family of t h e sequence s a t i s f y i n g t h i s propersy being

' t z g h t ' ,

i f B i s a compact s e t . See Billingsley [ I ] ) .

The most elaborate r e s u l t on compactness theorem for a family of r e a l p r o b a b i l i t y d i s t r i b u t i o n s i s t h e following, which being a s p e c i a l case of t h e w e l l known Prohorov theorem f o r a family of probability measures on ( S , S ) ,

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here S is a (separable and complete) metric space and

S

the class of Bore1 sets in S.

THEOREM

1.3. Let

{ X n }

(n=l, 2,. . .

⁾

be any sequence o f random uariabZes beZoizging t o F(R,B). Then Ix,} (n=1,2,. . .

⁾

i s relatively compact, that i s

{ X n }

[n=1,2,. . . contains a weakly convergent subsequence, if and only i f it has property B(S).

Next, we introduce

a

kind of uniform absolute continuity property of the sequence of random variables. The sequence

{ X n }

(n=1,2,. .

^.)

is said to have property

C ( C ) ,

if for any given

E > 0,

there exist a positive number 6 =

6(&)

and a positive integer N = N(E,~), such that

sup { p h ( E ) I

^{y ( B )}^<

^6,

^F^E^C.

ⁿ ^,N 1

^<^c.

This notion is necessary for the underlying convergence to be uniform onewhich will be discussed in the later section.

The purpose of this article is to generalize the above results to the case of complete compactness in the sense of some strictly stroger convrgence tRan the weak convergence. In Section

2,

some lemmas on asymptotic equivalence of real probability distributions are stated. In Section 3, extensionsof fore- going results are given.

2.

NECESSARY L M S

Let S* be the class of all subsets of

R of the form E* = {x =

(xl ,..., ^{xk) a,}

^{( x i}

⁵ ^B~ .

^i=l,2

^,.... ^,k

ai , Bi

:

extended rational

and let

A*

be the class of all finite disJoint unions of the members of **s*,**

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303

t h a t i s ,

COMPACTNESS THEOREMS

F i r s t , we s h a l l prove t h e following

LEMMA

2.1. If

one o f t h e sequences tXnl

( r s l , 2,.

. .

⁾

and {Yn}

(n=l, 2,.

. .

⁾

has property B ( S ) , then it holds t h a t

PROOF. Since S* C S, it i s r e a d i l y seen t h a t (( S))d--+ ((

s * ) ) ~ .

Therefore, it s u f f i c e s t o show t h a t (( s * ) ) ~ -3 (( S))d

.

Assume t h a t

X,

s

Yn

(( S* ))

( n

^{-t m ) ,} then it i s easy t o s e e t h a t both t h e sequences have t h e property B(S). Hence, for any given E > 0, t h e r e e x i s t a p o s i t i v e integer no and a s e t B belonging t o S, whose c l o s u r e being compact, such t h a t

'n ^Y

P (B) > 1

-

^E and P n ( ~ ) > 1

-

^E

,

f o r a l l n ,no For any s e t S belonging t o S, t h e s e t E defined by E = S

n

B i s bounded, and it holds t h a t

X 'n Y

Y

IP

^{n ( ~ )}

^-- ^P

^(El

I

^<^E ^and

IP

^{n ( ~ )}

^- ^P

^{n ( ~ )}

I

^<^E

^,

f o r a l l n 2 na

.

Note t h a t E belongs t o S, then for a l l n 'no, Y

~ b n ( s )

- / ~ ( s ) I

< l p X , ( ~ )

-

^P~ ( E ) I 2r.

Since for any s e t E belonging t o S t h e r e e x i s t s a monotone decreasing sequence of s e t s i n

s*,

say { E ~ J (n=1,2,. ..), such t h a t E: C_ B f o r each n and t h a t

ni=l

^E," ⁼E, then t h e r e e x i s t s a p o s i t i v e i n t e g e r m s a t i s f y i n g

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the following inequalities

X X Y 'n

IP ^"(E) - P "(E;)I

< c and

IP ^{n ( ~ )} ^- ^{P (E:)]}

^<^{c ,}

for all

n 2 m,

where E is the same value as above.. Thus, putting

2 =max(no,

m )

,

^{we have}

which implies that

Hence, it holds that

Xn

%

Yn ((S))d

^as

n

+

-,

which completes the proof.

In the similar manner to the above we can snow next results, whose proof will be omitted.

LEMMA 2.2. If one of the sequences {Xn} (n=l, 2,. .

⁾

and {Ynl (n=l, 2,.

,

.

⁾

has property B(S), then it holds that

(2

2 )

((A)& C- ((A))~* ..

Since

((A)& z ( ( S ) ) d

always holds as was shown in [2], we thus obtain the following

COROLLARY 2.1. If one o,f the sequence& {XnJ (n=1,2,. . .

⁾

and {Ynl (n=

1,2, . . .

⁾

has property

B (

S), then it

holds

that

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305

COMPACTNESS THEOREMS - 89

-

LEMMA 2.3. If a t Least one of the sequences {Xn} (n=1,2, ...) and

{Pn}

(n=1,2,.

..

^,) has property B(S) and C( S ) szmuZtaneousZy o r one f o r each, then z t hoZds t h a t

( 2 . 4 ) ( ( A ) ) d e

( M I d .

This lemma i s d i r e c t l y obtained by Theorem 3.2 i n

[2]

and t h e f a c t t h a t ((A)), i=t ((S)), d ( ( M J ) d holds without any condition.

3. EXENSIONS OF WEAK COMPACTNESS

THEOREMS

In t h e f i r s t place, we consider an extension of Helly's s e l e c t i o n theorem t o t h e c a s e of (( A*)& convergence

LEMMA 3

1. If a sequence of k-dzmenszonaj! random variabZes {Xn} (n=l, 2, ) has property B(S), then there exzst a subsequence {ml

g

^{{ n l} and a r e a l random varzabZe Y beZongzng t o F(R,B), the c7,ass of a l l 'proper' random varzabZes d e f i n e d o v e r (R,B), such t h a t

X

PROOF. Since 0

5

P n ( ~ )

5

1 (n=1,2,

.

⁾ for any subset E &

B

and t h e c l a s s S

*

c o n s i s t s of countable subsets of R , we can s e e , by using t h e so-

c a l l e d diagonal method, t h a t for some subsequence

{rn}

C_ in} t h e p r o b a b i l i t y

A *

measure P n ( ~ ) converges for every s e t E* belonging t o s*.

Let us put

$

f o r each E*

i n

^S*

.

We s h a l l now d e f i n e t h e s e t function Q ( A * ) Over A* by

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, * * * * *

where A* = $=1

Et ,

E; B

S

(z=1,

...,

^{N )}^and

^{E i n E}

^{= $}

( i f

j). Then, j

t h i s s e t function s a t i s f i e s t h e conditions t h a t

& ( R )

= 1 and 0 (

Q(A)*

( 1, and it i s f i n i t e l y a d d i t i v e over

s,*

a s i s e a s i i y shown.

Now, we s h a l l show t h a t t h e s e t functions defined above by ( 3 . 2 ) and

( 3

3) can be extended t o a p r o b a b i l i t y measure over

( R , B ) .

To t h i s end it i s r e - quired t o e x h i b i t t h a t ( z ) t h e c l a s s

B

i s t h e smallest o - f i e l d containing

A',

and

(ii)

Q i s o-additive over

A.*

Suppose t h a t

b be

t h e minimal o - f i e l d containing

A * ,

t h e n

A*

C

A

C

B.

?f we assume t h a t

B

C

B,*

then

A*

C

A

C

B

C

B.*

This c o n t r a d i c t s t h e f a c t t h a t

B*

i s t h e minimal 0 - f i e l d containing A*. Xence,

B*

C

B.

The r e v e r s e i n c l u s i o n r e l a t i o n can be s h ~ w n a s t h e following way. Any subset belonging t o

S

has t h e form of

E =

{ x =

( x x k ) a v ( x < b v , "=I, .... ^,k

av , bv

: extended real.

f+

" [ a l , bl)

x

[ a 2 , b 2 ) [ a k , b k )

tlnld we can chdbse monotone sequences of r a t i o n a l numbers; say

{ a v ,

t

, 16:)

and

{ y v ) (~"1,. . . ^,k;

^t=1,2,.

.

^.); such t h a t

a v , t B:, y:

a r e extended r a -

t t t t t t

Mona1 numbers,

av

4

a v , Bv + ^bv , yv + ^bv

^{a s} ^t^-t

^-,

^and

av

(

Bv

( yv f o r every

v

and t. P u t t i n g that, f o r every

i

=

1,2,. ..,M,

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307

COMPACTNESS THEOREMS -

91 -

Since an a r b i t r a r y subset A i n A i s represented by A =

liSl M

E z , where EZ (i=l,.

. .

^,M) ^E^S ^{with Ez}^~7^E

ti

⁼

+

( i # j ) , then for each Ez t h e r e e x i s t cer-,

*t * t

t a i n sequenpes of s e t s i n

**s*,**

say IEi } (t=1,2,

. .

^;^z=1,.

.,

^,M) ^and ^{G_{V Z}

. I

= 1 2; 1

. ^M

^{1 , .}

. . ^k

respectively, such t h a t

Hence A C B*. Noting t h a t B i s t h e minimal 0-field containing A, we ob- t a i n B C B*. Therefore, B* coincides with t h e Bore1 f i e l d

B,

which proves ( % )

We must now show (iC) ,. For every s e t E belonging t o

B ,

l e t

?

be an outer measure induced by t h e measure Q;

Then, f o r each i, choosing a covering of A: e A*, such t h a t

it follows from (3.4) t h a t f o r any given E

2

0 t A

m

*

M e ,

put E = A:, then E C

&ll;=l

^A;$ and t h a t

I

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Since Q i s an r e s t r i c t i o n o f

3

on A* and ^ED O is a r b i t r a r y , t h e n t h e above implies

Next, we must prove t h e contrary i n e q u a l i t y of

( 3 . 5 ) .

Noting t h a t t h e monotone non-decreasingness o f

V ( E ) ,

we g e t f o r any p o s i t i v e i n t e g e r N

The a r b i t r a r i n e s s o f N implies t h a t

* *

Hence, from (3.5) and

( 3 . 6 ) ,

it follows t h a t , for E = A: with Ai E A ,

<=1,2,

...,

which means t h a t Q i s o-additive over A

.

Therefore, t h e extension theorem assures us t h a t Q can be uniquely extended t o a p r o b a b i l i t y measure

&

over

B

by

( 3 . 4 ) .

(Cf. Kolmogorov [3] )..

We s h a l l denote a random v a r i a b l e corresponding t o t h i s p r o b a b i l i t y measure by

Y

It i s c l e a r t h a t

Y

belonging t o

F ( R , B ) ,

because property

B ( S )

of t h e sequence {Xn}(z=1,2

,...

⁾ i s brought over Y , too. P u t t i n g

g =

^,^'^P

we can now see t h a t

* *

f o r every s e t A belonging t o A

.

This comletes t h e proof o f t h e lemma..

Now, we a r e i n a p o s i t i o n t o s t a t e t h e following r e s u l t :

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3@9 COMPACTNESS THEOREMS -

93

-

THEOREM 3 . 1 .

Let CXn}

(n=1,2, "

.. .

⁾

be any sequence o f k-dimensions t rndom variables beZonging t o the class of aZl proper random varzables defined over (R,B). Then, there exzst a subsequence I m } C in1 and a real random varzabZe belonging t o

F(R,B),

such that

if and only

z f

z t has property B ( S )

PROOF. The " i f " p a r t immediately follows byLemnias 3 l a n d 2 2 As for t h e "only i f " a s s e r t i o n , suppose t h a t t h e sequence { X n } (n=1,2, . ) does not have property

B ( S )

Then, for some E. > 0, for any bounded subset

B

= B (

€ 1

belonging t o

S

and f o r any p o s i t i v e integer N =

N ( E , B ) ,

t h e r e e x i s t s a p o s i t i v e integer n , such t h a t

X

P ~ ( B ) ( 1 . - ^{E ,} for a l l

n z . ~ .

Define

BN

= 3:

1 ^-N 5 x$

< N, z=l,.. ,k } with

N =

[ I / € ] and put A N =

E ,-,

^BN for any subset E belonging t o A I f t h e r e e x i s t a subsequence

{m}

C In}

and a random v a r i a b l e

Y

belonging t o

F ( R , B )

such t h a t Xm + Y ((A))d

( n

⁺

"1,

then for every

m 2 N

^itwould follow t h a t

Y X

P

( A N )

= limm,

P

m ( ~ N )

5

lim-

P xm

( B N )

2

1

-

^{E .}

Especially taking t h e e n t i r e space

R

a s

E ,

it follows t h a t

p Y ( A N )

-+

p Y ( R )

< 1

-

^E ^{a s}

N

+

-,

which c o n t r a d i c t s t h e f a c t t h a t Y ^E

F ( R , B ) ,

and we com-

-

p l e t e t h e proof of t h e theorem.

Next, we s h a l l s t a t e a r e s u l t on t h e uniform convergent subsequence whose l i m i t i n g d i s t r i b y t i o n i s a b s o l u t e l y continuous with r e s p e c t t o t h e Euclid- Lebesgue measure

u.

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THEORE&i

3.2.

_{E d t}

{Xn3

(&I,

h, .'. .

⁾

be any' gcveh sequence o f k-dimensio- n a Z

random

variabZes beZo&ng to' h e ' e h s s*

of aZZ

ppoper d a r n variubles defiied over

(R,B).

fikn, there bx6s't h kuBsequetzce 12) E

{n3

and a reaz random variable

Y

belonging t o

p(R,B,p),

sttch that

provided that the sequence

IXnl (n=l, 2,.

. .

⁾ ^bps

proper6ies

B ( S )

and

c ( S )

simultaneous

Zy

.

PROOF. Since CXn} (n=1,2,.

. .

⁾ has property B ( S ) , by v i r t u e of preceding theorem, t h e r e e x i s t s a subsequence Im3

C In3

such t h a t

where

Y

i s a c e r t a i n random v a r i a b l e belonging t o F ( R , B ) . From t h e defini- t i o n of property C ( S ) of

{Xn)

(n=1,2,.

. .

^), t h e subsequence fXm3 has a l s o t h e property; f o r any given E > 0 t h e r e e x i s t a 6 = 6 ( E ) >

0

and

a

p o s i t i v e i n t e g e r

no

such Ohat f o r every subset A belonging t o A ,

implies t h a t f o r a l l m

2 n,

Thus, it follows t h a t f o r a l l

m 2 no

Since, by (3.101, bhe second term o r t.he XHS i n t h e above tends t o zero as

n

-> m

,

t h e n f o r s u f f i c i e n t l y l a r g e c mi

,

rc ) L 8

$ ( A ) <

,.

(3.13)

(14)

Now, f o r any subset E i n

B,

l e t {AZ} (2=1,2,.

.

⁾ ^be^BJcovering o f E consisting of mutually d i s j o i n t members of A, such t h a t P(Aik < & / p i and ,?'(A,) < E / 2' (i=1,2,

.

^), then by (3.11) it follows t h a t

and t h i s implies t h a t , v i a (3.131,

i

Therefore, Y belongs t o t h e family p(R,B,u), and hence (3.9) f o ~ l o w s by Lemma 2.3 and t h e well-known Polya theorem. Thus, t h e proof of t h e theorem

i s completed.

REkURKS: ( z ) It i s enough supposed t h a t Theorem 3.1 and Theorem 1.3 i s closely r e l a t e d But, out of accordance with our i n i t i a l purpose, up t o now t h e i n c l u - sion r e l a t i o n i s not necessarily clear between t h e r e l a t i v e l y compactness and t h e existence of (( A

)Id -

convergent subsequence In both theorems, anyway, we

,

see t h a t t h e assumption of propery

B(S)

i s e s s e n t i a l l y important f o r t h e under- l y i n g sequence t o be r e l a t i v e l y compact or completely compact. This property i s n a t u r a l l y understood a s a generalizednotioninstead of t h e conditon

u2,<

assumed i n Theorem 1.2.

( z i ) Under t h e condition of Theor em 3 . 2 (M)

, ^((MI)

((G))d holds,

d d

where G i s t h e c l a s s of a l l open subsets of R with respect t o t h e usual Euclidean distance. (Cf. [ 2 ] , [ k ] ) . Then, t h e convergence i n (3.9) becomes equivalent t o t h e we& convergence with a proper l i m i t i n g p r o b a b i l i t y d i s t r i - bution However, it seems t o t h e author t h a t t h i s theorem i s independently i n t e r e s t i n g and favorable t o know t h e s t r u c t u r e of uniform compactness based on t h e theory of asymptotic equivalence theory for r e a l p r o b a b i l i t y d i s t r i b u - t i o n s

(15)

(iii) Although we restricted our discussions to the case of fixed k- dimensional real probability spaces,it is of interest to extend our results to more general cases. But, it is still open whether the parallel versions in our notion to the results stated in Billingsley's book [l] are also valid or not.

Acknowledgment

The author wishes to thank Professor S. Ikeda for his valuable comments on an earlier draft.

Department of Information Science, College of Economics, Kagawa University

[l] Billingsley, P. (1968

1.

Convergence o f ProbabiZity Measures, John Wiley and Sons, Inc , New York.

121

Ikeda, S. (1968). Asymptotic equivalence of real probability distributions, Ann. I n s t . S t a t i s t . Math., 20, 339-416.

[31

Kolmogorov, A. N. (1950). Foundations o f Probabilz t y , Chelsea, New York .

141 Matsunawa, T, (1973). On inclusion relations for asymptotic equivalence of real probability distributions, Kagawa Eeo. Rev., 4 6 , 16-27.

COEdPACTNESS THEOREMS ON A REAL PROBABILITY SPACE