Limit Theorems for Proportions of Balls in a Generalized Urn Scheme

W O R K I N G P A P E R

L I M I T THEOREMS FOR PROPORTIONS OF BALLS I N A GENERALIZED URN SCHEME

W.B. A r t h u r Y u . M . E r m o l i e v Y u . M . K a n i o v s k i

O c t o b e r 1 9 8 7 WP-87-111

l n l e r n a l ~ o n a l l n s l ~ t u l e for Applied Syslerns Analys~s

Liait Theorems for Proportion8 of Balls _{in a}

Generalized U r n

Scheme

W. B. A r t h u r Y u . M. E r m o l i e v Y u . M. K a n i o v s k i

O c t o b e r 1 9 8 7 WP-87-111

Working P a p e r s a r e interim r e p o r t s on work of the International Institute f o r Applied Systems Analysis and have r e c e i v e d only limited review. Views or opinions e x p r e s s e d herein d o not necessarily r e p r e s e n t those of t h e Institute or of i t s National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 Laxenburg, Austria

In t h i s p a p e r t h e a u t h o r s continue to study t h e p r o c e s s of growth modeled by u r n schemes containing balls of d i f f e r e n t c o l o r s . The r a t e of c o n v e r g e n c e f o r p r o p o r t i o n s of balls t o t h e limit state i s investigated. I t i s shown t h a t Gaussian as well as non-Gaussian Markov random p r o c e s s e s may d e s c r i b e t h e asymptotic b e h a v i o r .

Alexander B. Kurzhanski Chairman

System a n d Decision S c i e n c e s P r o g r a m

Limit

Theorems for Proportions of

Balls

in a Generalized U r n Scheme

+ + + ++

W.B.

A r t h u r , Yu.M. E r m o l i e v a n d Yu.M. K a n i o v s k i

*

S t a n f o r d U n i v e r s i t y , 311 W. Encina Mall, CA, USA

**

Glushkov I n s t i t u t e of C y b e r n e t i c s , Kiev, USSR

1. Introduction

This p a p e r i s a c t u a l l y t h e s e c o n d p a r t of t h e p a p e r [ I 4 1 a n d f o r t h e s a k e of a b r i d g m e n t we a d o p t h e r e a l l n o t a t i o n s f r o m [ 1 4 ] . W e c o n s i d e r only t h e s i t u a t i o n when t h e v e c t o r X n , n +

-

of b a l l s p r o p o r t i o n s c o n v e r g e with p r o b a b i l i t y 1 t o a non-random v a r i a b l e . This will b e t h e case, in p a r t i c u l a r , when t h e set of f i x e d - p o i n t s of u r n f u n c t i o n s i s a s i n g l e t o n . The g e n e r a l case i s s t u d i e d in a f o r t h c o m i n g p a p e r .

We i n v e s t i g a t e a s y m p t o t i c n o r m a l i t y as well as t h e law of i t e r a t e d l o g a r i t h m a n d t h e i n v a r i a n c e p r i n c i p l e s . T h e s e r e s u l t s g e n e r a l i z e r e s u l t s of t h e a r t i c l e [ 8 ] . The main i d e a of o u r a p p r o a c h o r i g i n a t e s in p a p e r [ 3 ] a n d c o n s i s t s in a n i n t e r p r e - t a t i o n of r e l a t e d to t h e g e n e r a l i z e d u r n s c h e m e p r o c e s s e s as a s t o c h a s t i c a p p r o x i - mation t y p e p r o c e d u r e .

2. Auxiliary Results

I n s u b s e q u e n t t h e o r e m s a r e i m p o r t a n t l o c a l p r o p e r t i e s of t h e p r o c e s s Xn in a n e i g h b o r h o o d of a t t r a c t i n g p o i n t 9 . The following lemma e s t a b l i s h e s t h e n e c e s s a r y r e l a t i o n s .

Lemma 1. If f o r a n y a

>

0 a n d a l l z E u ~ - ~ ( ~ , E )

1 ) T h e r e i s r ² 3 s u c h t h a t s u p t a l ] i [' qt ( i ' 2 ) 5 C 1 , t h e n uniformly with i E Z ~

r e s p e c t to z , y

w h e r e

tt

i s t h e N-dimensional v e c t o r whose f i r s t N-1 c o o r d i n a t e s coin- c i d e with c o r r e s p o n d i n g c o o r d i n a t e s of v e c t o r q l a n d t h e l a s t c o o r d i n a t e i s ] f i t (Xt )[ - r t

(4 >.

L e t also

2) t h e r e e x i s t q 2 3 a n d v a r i a b l e s q ( i , z ) , z E

uN-l($,

^{E ) n LN}

-,,

^{i E Z ,} N

s u c h t h a t

x

q ( i , z ) = I ,

x

]iLq q ( i , z ) S C 3 ,

i €zf i €2:

( q ( i , z ) - q t ( i , z ) ) S ot - 0 with t ^{-+ a.} Then

w h e r e

a n d if. in addition.

3 ) f u n c t i o n s q ( i ^{, z ), i f}

Z?

are c o n t i n u o u s o n

uN-l('19,€)

n L N - i ; t h e n

c) u, f a n d p are c o n t i n u o u s o n this set of f u n c t i o n s . F u r t h e r m o r e , if c o n d i t i o n s Z ) , 3) are s a t i s f i e d and

4 ) p a r t i a l d e r i v a t i v e s q ( ' ) ( i . z )

= -.

^k

⁼

^1.2,

^....

N -1 e x i s t a n d are

a2

c o n t i n u o u s f o r all i €2:. z ^f u ' - ' ( I ~ , E ) n L N - j t a n d series

x

^{i j q ("(2 . z ) , j ,k}

⁼

^1,2,

^....

^{N -1} c o n v e r g e uniformly; t h e n

i €z$

d ) f a n d p a r e c o n t i n u o u s l y d i f f e r e n t i a b l e o n u " - ' ( 6 , ~ ) n L N - i a n d

If c o n d i t i o n s 1 )

-

3 ) a r e s a t i s f i e d a n d 5) Xt ⁺ I9 with p r o b a b i l i t y 1 f o r

t

+ =, t h e n

e ) with p r o b a b i l i t y 1 yt

t -'

⁺ p(I9) f o r t -, =.

Proof. F u r t h e r o n w e a s s u m e t h a t z E U ~ - ' ( + , E ) n LN-1 a n d y i s a n a t u r a l n u m b e r . Using e s t i m a t e (7) f r o m [ I ] , c o n d i t i o n 1 ) a n d H o l d e r i n e q u a l i t y w e o b t a i n f o r X t = z , y t = y

If R

>

2 ~ , 1 / ' ( 4 N

+

1)'12 t h e n

llltt

(Xt , y t

)[I2 >

R 2 j L 12(4N+1)(lPt

(xt )r2 ⁺

a n d , t a k i n g i n t o a c c o u n t i n e q u a l i t y ( 1 2 ) f r o m [ I ] with p = r , p

=

2 a n d p

=

0 w e ob- t a i n

Thus, t h e a s s e r t i o n 2 ) i s p r o v e d .

If t h e c o n d i t i o n 2 ) i s s a t i s f i e d , t h e n o n t h e b a s i s of e s t i m a t e (8) f r o m [I] a n d H o l d e r i n e q u a l i t y

T h e r e f o r e d u e to a s s e r t i o n s a ) , b ) of Lemma 1 f r o m [I]

w h e r e j ,k = 1 , 2 ,..., N -1. I t i s e a s y to see t h a t

Z + T

S i n c e - 5 2 with T t 0 t h e n f r o m c o n d i t i o n I ) , e s t i m a t e (8) f r o m [I] a n d t h e 1 +T

H o l d e r i n e q u a l i t y

A s i n p r o v i n g a s s e r t i o n b ) of Lemma 1 in [I], ta k i n g i n t o a c c o u n t (8) a n d (12) with p

=

m i n ( t ,q ), p

=

2 f r o m [I] a n d c o n d i t i o n s I ) , 2 )

- 5 -

Assume t h a t L = L ( t )

=

o t - ' l N tH, t h e n

S i n c e

t h e n b y r e a s o n i n g in t h e s a m e m a n n e r as when p r o v i n g a s s e r t i o n of Lemma 3 f r o m [I] w e o b t a i n

Based o n c o n d i t i o n s I), 2 )

t h e r e f o r e , d u e to ( 6 )

Taking i n t o a c c o u n t c o n d i t i o n s I), 2), a s s e r t i o n s a ) , b ) of Lemma 1 fr o m [I) a n d ( I ) , (7)

w h e r e j

=

1 , 2 ,

...,

N-1. From d e f i n i t i o n

t t ,

r e l a t i o n s (2) - (12) a n d t h e f a c t t h a t by tH

=

0[otm] f o r t ^-+ t h e v i r t u e of c o n d i t i o n 2 ) ot -+ 0, a n d t h e r e f o r e , ot

a s s e r t i o n b ) follows.

From 2 ) , e s t i m a t e s (8) a n d (12) with p = q , p

=

2 o r p

=

1 f r o m [I], s e r i e s which d e f i n e f u n c t i o n s o , f a n d p c o n v e r g e uniformly o n u N - ' ( 3 , ~ ) n LN-1. Then i t s sum i s a c o n t i n u o s f u n c t i o n [9, p . 4311, a n d u n d e r c o n d i t i o n 3 ) t h e a s s e r t i o n c ) i s valid. By d i f f e r e n t i a t i n g f o r m a l l y t h e e x p r e s s i o n f o r f j w e o b t a i n

w h e r e djk i s a K r o n e c k e r symbol. On t h e b a s i s of a s s e r t i o n c ) t h e f i r s t t e r m h e r e i s a c o n t i n u o u s f u n c t i o n , a n d with a c c o u n t f o r c o n d i t i o n 4 ) t h e s e r i e s c o n s i s t s of c o n t i n u o u s f u n c t i o n s a n d c o n v e r g e n c e s uniformly. T h e r e f o r e [9, p. 4311 ^{a f j} ( x

a x k ' j , k , = 1 , 2 ,

...,

N-1 e x i s t a n d are c o n t i n u o u s on

uN

- l ( d , s ) n LN-i a n d f i s continu- o u s l y d i f f e r e n t i a b l e o n t h i s s e t . From e q u a l i t y (13) i t follows t h a t

i . e . , with a c c o u n t f o r t h e f o r e g o i n g , p i s continuously d i f f e r e n t i a b l e o n UN -I(+, E ) n LN a n d t h e a s s e r t i o n d ) i s valid.

Based on r e l a t i o n s (17), (18) f r o m [I] to p r o v e t h e a s s e r t i o n e ) i t s u f f i c e s to show t h a t with p r o b a b i l i t y 1

n ^-1

n - I ri(Xi) ^-+ p(I9) f o r n + =

This r e l a t i o n i s valid by (7) a n d t h e c o n t i n u i t y of t h e f u n c t i o n p ( a c c o r d i n g to c ) ) o n UN-l(I9,e) n LN-l. T h e r e f o r e , t a k i n g i n t o a c c o u n t c o n d i t i o n 5) p(Xt) + p(I9) with p r o b a b i l i t y 1 f o r t ⁺ =. The lemma i s p r o v e d .

Remark 1. If N = 2 , f u n c t i o n s q ( i , x ) , i € 2 ; are c o n t i n u o u s o n t h e set (I9

-

^{E ,} I9

+

^{e )} nR ( 0 , l ) ; s e r i e s

C

^{( i l}

+

^{i 2 ) q} ( i , z ) c o n v e r g e s uniformly; o n t h e set

i E Z , ~

(I9 -e, 19) n R ( 0 , l ) or ($,I9 + e ) n R ( 0 . 1 ) f u n c t i o n s q l ( i , z ) e x i s t , are c o n t i n u o u s a n d s e r i e s i j q l ( i , x ) , j

=

1 , 2 c o n v e r g e uniformly, t h e n s i m i l a r to p r o o f of asser-

i €2,2

t i o n d ) i t c a n b e shown t h a t t h e f u n c t i o n f i s c o n t i n u o u s o n (I9

-

e , I9

+

e ) n R ( 0 , l ) a n d i s c o n t i n u o u s l y d i f f e r e n t i a b l e o n (I9 -&, 19) n R(O,1) o r ($,I9

+

^e) n R ( 0 , l ) r e s p e c t i v e l y .

The following facts a r e useful to s t u d y t h e asymptotic of t h e u r n p r o c e s s e s . L e t M dimensional v e c t o r s z , , s 2 1 form a Markov p r o c e s s ,

z, -, 3 with p r o b a b i l i t y 1 f o r s -, = , ( 1 5 ) where r

>

0 , 3 E R M . Assume t h a t t h e r e i s E

>

0 s u c h t h a t f o r z , = z , z ^E u M ( d , r ) , s 2 I

max (s 112: -3))) -, 0 ,

w h e r e g and w , are M-dimensional vector-functions; D ( s ,z )

=

E z ( s , z ) z ( s , z ) T , D

-

i s a symmetric non-negative d e f i n i t e m a t r i x ,

Il.)lo

i s a norm of M x M m a t r i c e s . I n all s u b s e q u e n t lemmas r e l a t i o n s ( 1 4 ) - ( 1 8 ) are assumed to b e s a t i s f i e d .

L e t D M [ O , ~ ] b e a s p a c e of M-dimensional v e c t o r functions on [ 0 , T I , T

>

⁰

without s e c o n d o r d e r discontinuities with S k o r o k h o d m e t r i c

[ l o ]

( f o r M

=

1 simply D[O. T I ) . In D M [ o , T I and D [ 0 , T I f o r n 2 2 we c o n s i d e r random p r o c e s s e s

n +S n +s +1

X n ( t ) = ( n + ~ ) ~ / ~ ( z , + ~ - d ) f o r

x

^{i - I ?S t}

^< C

^{i - I ,}

and

Lemma 2. 151. Assume t h a t

1 ) in e q u a l i t y ( 1 6 ) function g i s d i f f e r e n t i a b l e at point 19, i.e., f o r z

-.

I 9 , g ( z )

=

G(z-13)

+

o(lb+I();

2 ) m a t r i x G

+

-JM i s s t a b l e (i.e., r e a l p a r t s of i t s eigen-values 1 are n e g a t i v e ) 2

w h e r e JM is a unit m a t r i x i n R M ;

l i m ~ l ' ~ sup ~ ~ W ~ ( X ) ( ( = O

.

s +- z € U M ( ~ , C )

Then with n ^-r r a n d o m p r o c e s s e s

X,

weakly c o n v e r g e in D ~ [ o , T ] to a s t a t i o n a r y g a u s s i a n Markov p r o c e s s X, which s a t i s f i e s a s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n of t h e following f o r m

w h e r e 0'" i s a non-negative s q u a r e root of t h e s y m m e t r i c m e t r i x

D ,

w~ i s a s t a n - d a r d M-dimensional Wiener p r o c e s s (with M = 1 , simply w ) .

Lemma 3. L e t M = 1 a n d 1 ) with x -r 79

g ( x ) = - I ( x -79)

+

⁰ ((x - - + I ) , 2

lim ( ~ l n s ) ' / ~

s ^{+ -} k s ~ ~ e I " s ( " ) I = 0

.

Then with n ^{- r -} r a n d o m p r o c e s s e s Y, weakly c o n v e r g e in D[O,T] to a s t a t i o n a r y g a u s s i a n Markov p r o c e s s Y, s a t i s f y i n g t h e following s t o c h a s t i c d i f f e r e n t i a l e q u a - t i o n

The p r o o f of t h i s lemma i s b a s e d o n limit t h e o r e m s f o r r a n d o m p r o c e s s e s gen- e r a t e d b y s e r i e s of weakly d e p e n d e n t r a n d o m v e c t o r [ll] a n d i s s i m i l a r to t h a t in P a p e r

P I .

Lemma 4. [4]. L e t M

=

1, 1 ) f o r x ^-r 19

Then with p r o b a b i l i t y 1

-

[

ⁿ

j1'2[-~;~~j1'2

lim

-

( 5 , - 9 ) = 1 ,

n +.. LnLnn

Lemma 5 [6]. S u p p o s e t h a t M = 2 , 1) with x -, 9

g ( s ) = G ( s 1 ) ( s - . 9 ) +o(lk-L911> ^# w h e r e G ( s l ) = G1 f o r z i 2 d l a n d ~ ( 5 ' ) = G 2 f o r s1

<

d l ;

2 ) m a t r i c e s Gi

+

-J2, I i

=

1 , 2 a r e s t a b l e ; a n d c o n d i t i o n 3) of Lemma 2 i s s a t i s - 2

f i e d .

Then with n -, ~0 r a n d o m p r o c e s s e s

&

weakly c o n v e r g e s in D ~ [ o , T ] to a sta- t i o n a r y Markov p r o c e s s X which s a t i s f i e s a s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n of t h e following f o r m

Now we c a n p r o v e some r e s u l t s o n a s y m p t o t i c b e h a v i o r of b a l l s p r o p o r t i o n s in t h e g e n e r a l i z e d u r n s c h e m e .

3. Limit Theorems

In D*[o, T I , T

>

0 , we c o n s i d e r r a n d o m p r o c e s s e s

w h e r e n 2 1 , y n , G are N-dimensional v e c t o r s whose N -1 c o o r d i n a t e s are e q u a l c o r r e s p o n d i n g c o o r d i n a t e s of t h e v e c t o r s X n , 9 , a n d t h e l a s t c o o r d i n a t e s are k v ~ ( 1 9 ) v k

=

7 n / n .

T h e o r e m 1. L e t

I ) s u p s u p

2 ^{l i}

^{Cqn ( i ,z}

^{= el;}

rial z ~ L -

E ~ t

I ~

2 ) X, + 19 with p r o b a b i l i t y f o r n -, a n d f o r s o m e E

>

0 t h e following c o n d i t i o n s

hold on uN-'(-19, E ) n LN

3 ) t h e r e e x i s t s r L E f o r which s u p

z

^Ji [ ' q , ( i , z )

=

C Z ; i E Z ~

4 ) t h e r e e x i s t s c o n t i n u o u s l y d i f f e r e n t i a b l e f u n c t i o n s q ( i , z ), i ^E Z? s u c h t h a t

z

q ( i , z ) = 1 f o r s o m e P 2 3 ,

z

1 i I q q ( i , z ) 5 Cg.

i

€zf

^{i EZ$}

a

^{z z}

l q ( i . z ) - q n ( i . z ) l d o n . n L l , s e r i e s

z

i j U n j , k = 1 . Z ,.... N - 1 , con-

i EZ+ axk

v e r g e uniformly;

n N ^1.H 5 ) lim an- = 0 ;

n ^+- Z H

6) m a t r i x A

+

-JN 1 i s s t a b l e , w h e r e ~ j ~ = p ( 9 ) - l a , A j = ~0 , j = 1 . 2 ,..., N - 1 ,

z axk

Then r a n d o m p r o c e s s e s zn weakly c o n v e r g e in f l [ o , i " ] to a s t a t i o n a r y gaus- sian Markov p r o c e s s z which s a t i s f i e s t h e following s t o c h a s t i c d i f f e r e n t i a l e q u a - t i o n

w h e r e

C i j ( 9 )

=

p(19) -'aij ( + ) , i

=

1 , Z , .

. .

, N -1 ,

f l j ( I 9 ) = p ( I 9 ) - ' & ( I 9 ) , j = 1 , 2 , ..., N - 1 , f l N ( 1 9 ) = P N ( I 9 ) + F ~ ~ ( I ~ )

.

Proof. On t h e b a s i s of e q u a l i t y ( 5 ) f r o m [ I ] w e h a v e

t h e r e f o r e u n d e r c o n d i t i o n 1 )

S i n c e X: E ( 0 , 1 ) , i

=

1 , 2 ,..., N -1, n r 1, t h e n d u e to (20), (21)

On t h e b a s i s of a s s e r t i o n e ) of Lemma 1 a n d c o n d i t i o n s 2 ) , 5 ) a r e s a t i s f i e d w e h a v e pn -, p ( 9 ) with p r o b a b i l i t y 1 , n -,

-

^,

a n d , t h e r e f o r e

y n ^-+ 9 with p r o b a b i l i t y 1 , n

-

^-+

-

^{, (23)}

L e t eO

=

m i n ( e , p ( 9 ) / 2 ) . D e n o t e by yn ( 2 , ~ ~ ) a set of p o i n t s y ^E u N ( 5 , e O ) whose f i r s t N -1 c o o r d i n a t e s z ( y ) belong t o

uN

-I(+, e ) n LN a n d t h e last y N i s s u c h t h a t n y N is a n a t u r a l n u m b e r g r e a t e r or e q u a l yl. Using e q u a l i t i e s (6) f r o m [I], (19), a s s e r t i o n s a ) , b ) of Lemma 1 f r o m [I], a n d a s s e r t i o n b ) of Lemma 1 with c o n d i t i o n s 3 ) , 4 ) w e h a v e f o r yn

=

y , y E Yn

(3,

eo), n 2 1

w h e r e

Due to c o n d i t i o n 4), a s s e r t i o n d) of Lemma 1 a n d e q u a l i t i e s (27) t h e f u n c t i o n R i s d i f f e r e n t i a b l e at p o i n t

5.

S i n c e f(19)

=

0, t h e n

f o r

cn

⁺

^4,

^{y", E} Y ~ ( ~ , E ~ ) . From e s t i m a t e ( I ) , c o n d i t i o n s 3 ) , 4) a n d a s s e r t i o n s a ) , b ) of Lemma 1 we h a v e

Since f o r

g,

^-, w e h a v e x ( g n ) ^-+ 19 a n d t h e functions o,f , p are continuous on

u N

-'(19,c) n LN-l from condition 4 ) a n d a s s e r t i o n c ) of Lemma 1, th e n d u e t o condi- tion 5 ) w e o b t a i n

lim ~z k ( n , < , ) z j ( n , c n )

-

^Ckj(9)1

=

0 , max ( n -l,IIcn

- 211)

^-+ 0

where

Relations (22)-(26), (28)-(30) and conditions 5 ) , 6 ) make i t possible t o use lem- ma 2 which g i v e s t h e r e q u i r e d r e s u l t . The theorem i s p r o v e d .

Calculating d i r e c t l y t h e limit distribution of z ( t ) with t ^{-+ m} we o b t a i n t h e fol- lowing a s s e r t i o n .

C o r o l l a r y 1. Under conditions of Theorem 3

in p r o b a b i l i t y , where N ( 0 , B ) i s a normal random v e c t o r of N dimensionality with z e r o mean a n d v a r i a n c e matrix

Remark 2. Let t h e number of balls added t o t h e u r n at e a c h s t e p b e c o n s t a n t and e q u a l t o V 2 1 ( a s , e.g., in [3], [7], [ 8 ] ) . i.e.

Then conditions I ) , 3) of Theorem 1 are s a t i s f i e d , 7,

=

y l

+

( n - l ) v , n t 1, i.e., t h i s i s not a random v a r i a b l e and instead of z, i t i s s u f f i c i e n t t o c o n s i d e r t h e c o r r e s p o n d i n g N-1-dimensional random p r o c e s s g e n e r a t e d by X,,

-

9, n 2 1.

From r e l a t i o n s (22)-(26), (28)-(30), Lemma 3.4 a n d R e m a r k 1.2 of t h e p a p e r [ I ] w e o b t a i n t h e following.

T h e o r e m 2. L e t N

=

2 a n d t h e n u m b e r of b a l l s a d d e d to t h e u r n a t e a c h s t e p i s e q u a l t o v c o n s t a n t v 2 1 . S u p p o s e a l s o t h a t t h e following c o n d i t i o n s are s a t i s - fied:

1 ) with p r o b a b i l i t y 1 Xn ⁺ 19, n ⁺ -;

2 ) f o r some r

>

0 , x E (19-E, I9

+

^{E )} n R ( 0 , l ) t h e r e e x i s t c o n t i n u o u s l y d i f f e r e n t i - a b l e f u n c t i o n s q ( ( i , v - i ) T , x ) , ~

s

i

s

v , s u c h t h a t I q ( ( i , ~ - i ) ~ , x )

-

q n ( ( i , v - i ) , x ) l S o n , T n 2 1 ;

f"(I9)

=

- 1 / 2 , lim ( n ~ n n ) ' / ~ u ~ = O

.

n ^+- Then r a n d o m p r o c e s s e s

1/ 2

n + s n +S

( X n + , - 19) f o r

x

( i L n i ) - I S t =n

c o n v e r g e in D[O,T] f o r n ⁺

-

^to a s t a t i o n a r y g a u s s i a n Markov p r o c e s s v of t h e following f o r m

w h e r e

C o r o l l a r y 2. If c o n d i t i o n s of T h e o r e m 2 are s a t i s f i e d , t h e n

i n p r o b a b i l i t y f o r n ⁺

-.

T h e o r e m 3. L e t a l l c o n d i t i o n s of T h e o r e m 2 b e s a t i s f i e d e x c e p t t h e 3 ) which i s r e p l a c e d by t h e following

[

^j112Ds

⁼

⁰

^.

f ' ( 1 9 )

<

-11 2 , lim -

+- LnLns Then

-

_u

lim

d- ^(xn

^-19)

⁼

d - l d ( 1 9 ) ,

n +inf lnlnn -1 -2

In T h e o r e m s 1,2 t h e limit random p r o c e s s e s a r e g a u s s i a n , a n d t h e limit d i s t r i - bution of v a r i a b l e s

X,

¹⁹ i s n o r m a l . I t a p p e a r s t h a t if w e d i s c a r d t h e r e q u i r e m e n t of d i f f e r e n t i a b i l i t y of f u n c t i o n s q ( i ; ) , i E Z f , a t p o i n t I9 t h e n t h e limit random p r o c e s s e s may n o t b e g a u s s i a n , as well as t h e limit d i s t r i b u t i o n of v a r i a b l e s Xn-I9 n o t b e unfinite divisible. T h e t h e o r e m g i v e n below s t i p u l a t e s t h e a p p r o p r i a t e r e s u l t . The p r o o f of t h i s r e s u l i s b a s e d o n r e l a t i o n s (22)-(26), ( 2 8 ) - ( 3 0 ) , Remark 1 a n d Lemma 5 .

Theorem 4. L e t N

=

2,29 E ( 0 , 1 ) , c o n d i t i o n s 1-3, 5 of T h e o r e m 1 b e s a t i s f i e d as well as t h e following

4 ) t h e r e a r e c o n t i n u o u s l y d i f f e r e n t i a b l e o n (19-c, 19) n R ( O , l ) , (29,29+c) n R ( 0 , l ) f u n c t i o n s q ( i ; ) , i E ~ f s u c h t h a t

x

q ( i , x ) = 1, f o r some

i E Z ?

5 ) m a t r i c e s Ai C Z J 2 1 a r e s t a b l e , w h e r e i

=

1.2, A:' = p ( ~ ) - 1 j ' ( 1 9 + ~ ) ,

Then r a n d o m p r o c e s s e s zn weakly c o n v e r g e i n D ~ [ o ,

TI

to a s t a t i o n a r y M a r k o v p r o c e s s z , s a t i s f y i n g t h e following s t o c h a s t i c d i f f e r e n t i a b l e e q u a t i o n

w h e r e A ( z l )

=

Al f o r z 1 h O , A ( z l ) = A 2 f o r z 1

<

0 .

Corollary 3. L e t N

=

2 , d E ( 0 , l ) a n d t h e n u m b e r of b a l l s a d d e d to t h e u r n at e a c h s t e p b e c o n s t a n t a n d e q u a l to u 2 1.

S u p p o s e t h a t

1) z, -, 19 with p r o b a b i l i t y 1 , n ^-, =;

2 ) f o r some E

>

0 t h e r e e x i s t c o n t i n u o u s l y d i f f e r e n t i a b l e o n (19-&,19) a n d (19,19+&) f u n c t i o n s q ( ( i , v - i ) T , x ) , O 6 i ^{6 u} s u c h t h a t

maxCp'(d+O),f'(19-0))

<

- 1 / 2 , lim ni'20n = O

.

n +-

Then t h e limit d i s t r i b u t i o n of r a n d o m v a r i a b l e s dn(X,-19) h a s t h e d e n s i t y of t h e following f o r m

w

w h e r e C i s a c o n s t a n t s u c h t h a t J p ^(I ) d z

=

1.

-00

C o r o l l a r y 3 follows f r o m T h e o r e m 4 , R e m a r k 2 a n d t h e f a c t t h a t t h e limit dis- t r i b u t i o n z ( t ) ,

t

-,

-

h a s t h e d e n s i t y p . The d i s t r i b u t i o n with d e n s i t y p i s n o t un- f i n i t e d i v i s i b l e .

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[Ill

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