9. M . A . KrasnosePski i e t a l . , Integral Operators in Spaces of Summable Functions [in Russian], Nauka, Moscow (1966).

10. M . Z . Solomyak, "Anatyticity of semigroups generated by an elliptic opera tor in Lp spaces," Doki. Akad. Nauk SSSR, 127, No. 1, 37-39 (1959).

11. S . M . Nikol'skii, Approximation of Functions of Several Variables and Imbedding Theorems [in Russian], Nauka, Moscow (1969).

12. V . A . Solonnikov, "On general boundary problems for sys tems of differential equations of elliptic and parabolic type," in: Proceedings of the Joint S o v i e t - A m e r i c a n Symposium on Par t ia l Differential Equations, Novosibirsk [in Russian], Izd. Akad. Nauk SSSR, Moscow (1963), pp. 246-253.

M A T H E M A T I C A L E X P E C T A T I O N OF C O N T I N U O U S F U N C T I O N S

OF R A N D O M V A R I A B L E S . S M O O T H N E S S AND V A R I A N C E

L . 1. S t r u k o v a n d A. F . T i m a n UDC 517.5

1. Let ~ be a random variable and E(g) and D(~) its expectation and var iance, respect ively. The l inear functions f(x) = ax +/3 can be charac te r ized as the functions continuous on the entire real line, far which

Z:'[/(~) ] --i[E(~) 1, (1)

whatever the random variable ~. If the function fix) is not l inear, the difference E [f(~)] - f[E(~)] is nonzero and depends on the function f and the random variable ~.

This paper is devoted to the problem of est imating this difference in generally accepted probabilist ic and function-theoret ic t e rms . It is established that the change in the value E [f($)] when the order of the opera - tions of E and f is changed is co r rec t ly descr ibed by a natural and general inequality, which has hitherto not been noticed in the l i terature, and which expresses the dependence of this change on the degree of approxima- tion of the function f(x) by l inear functions and on the var iance of the random variable ~. It is shown along with this that other resul ts sharpening previously known es t imates for approximation of a function by Bernstein polynomials are also valid in the case of a binomial distribution law, when we consider in the role of fix) c e r - tain typical functions with s ingulari t ies .

2. We f i rs t state the asser t ions to be established here.

THEOREM 1. on the real line R for which

For any random variable ~ with finite var iance, if f(x) is any continuous function defined

o)~([;t)== s u p ](xl)--2][ zl--z2~ <

we always have the inequality

I E [] (~)] - - ] [E (~)11~ 3(o~ {]; U., F D (~)}.

If the function f(x) is defined only on some closed interval I of the real axis 1~ and

I ~--xd~2l Xl, X2~1

then for cer ta in values of the constants a and/3, the difference f(x) - ax - f~ can always be extended to R in such a way that [1, p. 135]

(2)

(3)

Trans la ted from Sibirskii Mate maticheskii Zhurnal, Vol. 18, No. 3, pp. 658-664, May-June, 1977. Original ar t ic le submitted Februa ry 6, 1975.

This material is protected by copyright registered in the name of Plenum Publishing Corporation, 227 blest 17th Street New York N, Y. I001L No part I of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, !

[ microfilming, recording or otherwise, without written permission o f the publisher. A copy of this article is available from the publisher for $ ZSO. j

469

In a l t such c a s e s , Eq. (3) l eads to the e s t i m a t e

IE l/(~)] -- ] [E (~)11 ~ t5r [1; (4)

for the extended function. This estimate improves a recent result of [2].

In concrete situations, the general inequality (4) contains corresponding estimates for approximations of functions f(x) on I which take into account the location of x~I. Among these, we remark, e.g., the estimate

- [o l ] I ] (x) 1 ~< ~a~e' {/; ~I,: Vx ( i - x)} (5)

f o r the va lues of If(x) l a t the points x ~ [ 0 , 1] unde r the condi t ion f(1) = f(0) = 0, which s h a r p e n s one of the r e - su l t s of Whitney [3]. We r e m a r k a l so the inequal i ty

I i (x) - -B~ (1; x ) [ ~ i5(,)~~ (]; 110]/ :~ (l --~)1 _ . _~ f (6)

contained in (4) for the approximations by Bernstein polynomials on the interval [0, 1] [4], which corresponds to the frequency } = }n of some event for n independent tests in a Bernoulli scheme; we also remark the esti- mate

X h .

i (z) - e --~ ~ ! (k) ~ ~ 15 ~o~,~ (L '/~. VT) (7) h=0

on the semiaxis 0 - x < ~o if ~ has a Poisson distribution. For random variables ~ with uniform or normal distribution, probabilities (3) lead to estimates of approximations by Steklov polynomials and singular Weierstrass integrals, in these two cases, the right-hand side of (3) does not depend onx~R and is always weakly equivalent to a corresponding uniform approximation of the function f(x) under study in the metric of the space C on (_0% ~) (cf. [5; 6, p. 501]). The situation is different for Eqs. (5)-(7), where the right-hand side as well as the left already depends onx, if we compare both sides of these equations, taking into account the location of the point x~I. In the general case, the order of the estimates just mentioned cannot be im- proved. At the same time, a consideration of the function xlnx shows at once that for random variables cor- responding to (5)-(7), weak equivalence between the two sides of these equations can fail even on some subse- quence of points tending to an endpoint of I. Such a situation is obtained for random variables in a Bernoulli scheme for all values of n. In this sense, a more precise estimate for the approximation of the function xlnx by Bernstein polynomials is given by the following assertion.

THEOREM 2. For any value of the positive integer n, if

B~(x) B ~ ( I ; x ) = = T k x ~ ( t -

i s the dev ia t ion of the funct ion x l n x f r o m i ts a p p r o x i m a t i n g B e r n s t e i n po lynomia l s , we have the inequal i ty

O ~ x in x-l--B,~(x) ~ n - ' [ l--q)~ (x) --%, ( l - -x ) J (8)

holding e v e r y w h e r e on the i n t e r v a l 0 -< x -< 1, w h e r e

t n

0

(9)

F o r each f ixed va lue of a, the r i g h t - h a n d side of (8) when x -* 0 has o r d e r of decay to z e r o coincid ing with the t rue o r d e r of decay of the dev ia t ion in the above t h e o r e m , and hence the r i gh t -hand side of (8) g ives an e s t i m a t e unob ta inab le f r o m the g e n e r a l inequal i ty (6). We r e m a r k a l so the inequal i ty

max l x lnx- l - -Bn(x) I ~ 1 / n (10) 0</x~<l

r e s u l t i n g f r o m (8), which i m p r o v e s s o m e w h a t a r e s u l t r e c e n t l y obta ined in [4]. Us ing the me thod of p roof of T h e o r e m 2, o t h e r e s t i m a t e s m o r e p r e c i s e than (4) can be ob ta ined fo r a p p r o x i m a t i o n by B e r n s t e i n po lynomia l s of a funct ion f(x) having s i n g u l a r i t i e s a t the end -po in t s of the i n t e rva l [0, 1]. Thus , e . g . , we have the next

t h e o r e m .

T H E O R E M 3. T h e p r e c i s e o r d e r r e l a t i on

max Ix~--B.(x) l x l/n: ( n f l ) (11) 0~x~<l

470

h o l d s fo r the d e v i a t i o n of the func t ion f(x) = x a (0 < (~ < 1) f r o m i t s B e r n s t e i n p o l y n o m i a l s .

Kac [7] has s t u d i e d the u n i f o r m a p p r o x i m a t i o n of the func t ion I x , 1/2 I s (0 < o~ < 1) b y B e r n s t e i n po'Ly- n o m i a l s and h a s shown tha t in th is c a s e the o r d e r of a p p r o x i m a t i o n n - a /2 is exac t . T h i s f ac t t o g e t h e r wi th (11) r e f l e c t s the in f luence tha t the d i s t r i b u t i o n of s i n g u l a r i t i e s of a func t ion h a s on the o r d e r of i t s a p p r o x i m a t i o n b y B e r n s t e i n p o l y n o m i a l s . A s is a l s o t r u e fo r b e s t u n i f o r m a p p r o x i m a t i o n s , in the c a s e of a p p r o x i m a t i o n by B e r n s t e i n p o l y n o m i a l s of i nd iv idua l func t ions hav ing s i n g u l a r i t i e s , the u n i f o r m a p p r o x i m a t i o n e s t i m a t e depends not only on t h e d e g r e e of s m o o t h n e s s of the func t ion , but a l s o on the d i s t r i b u t i o n of i t s s i n g u l a r i t i e s . Thus , e . g . , we can show by c o n s i d e r i n g the func t ion f(x) = Ix - cl (0 < c < !) tha t

m ~ l I x - ~ I - ~ . ( ~ ) l = o [ V ~ l - ~)~], (~ 2) O ~ x ~ i

w h e r e 0(1) is a quan t i t y u n i f o r m l y bounded fo r a l l c (0 < c < 1) and n.

In connec t ion with e s t i m a t e (11) showing tha t the o r d e r of u n i f o r m a p p r o x i m a t i o n on [O, 1 ] of fl~e func t ion x ~ fo r 0 < ot < t i m p r o v e s a s a i n c r e a s e s , i t is of i n t e r e s t to o b s e r v e the fo l lowing r e s u l t of a d i f f e r e n t n a t u r e r e l a t i n g to the s a m e func t ion fo r i < a < 2.

T H E O R E M 4. F o r e v e r y a (1 < o~ < 2) and p o s i t i v e i n t e g e r n, the i nequa l i t y

O ~ B , ~ ( x ) - - x ~ [ ( a - - l ) in] [ l - - q ~ (x) --(p~ ( i - - x ) ]

!s va l id fo r the d e v i a t i o n of the func t ion x ~ f r o m i t s B e r n s t e i n p o l y n o m i a l s e v e r y w h e r e on the i n t e r v a l 0 <- x -< 1, w h e r e the func t ions (Pn(X) a r e de f ined by (9).

We r e m a r k tha t by the w e l l - k n o w n l i m i t t h e o r e m of V o r o n o v s k a y a , and in the c a s e when 0 < c~ < 1, the o r d e r of a p p r o x i m a t i o n of the funct ion x c~ by the p o l y n o m i a l s Bn(x) is equa l to 1 /n at each f ixed point of the i n t e r v a l (0, 1). H o w e v e r , th is o r d e r d i f f e r s e s s e n t i a l l y f r o m the o r d e r of i t s u n i f o r m a p p r o x i m a t i o n which is equa l to 1 /n% If ce >_ 2, then the e s t i m a t e

max [x=--B, ,(x)}=O(t/n) 0 ~ < x < ~ l

f o r the u n i f o r m a p p r o x i m a t i o n is a s i m p l e c o n s e q u e n c e of the w e l l - k n o w n inequa l i t y

t j (x ) - -B , , ( / ; x) I~< max ] ] " ( t ) I x ( i - - x ) / 2 n , O ~ < t ~ i

and, t h e r e f o r e , fo r such v a l u e s of a i t is of a t r i v i a l c h a r a c t e r . The s i t u a t i o n is d i f f e r e n t in the c a s e when 1 < a < 2. Now, a s in the c a s e 0 < a < 1, the s e c o n d d e r i v a t i v e of x c~ b e c o m e s unbounded on the i n t e r v a l (0, 1). The above c o n s i d e r a t i o n s f o r ob t a in ing u n i f o r m a p p r o x i m a t i o n e s t i m a t e s equa l to O(1/n) and con t a ined in T h e o - r e m 4, a r e no l o n g e r a p p l i c a b l e , and the g e n e r a l t h e o r e m s on a p p r o x i m a t i o n by B e r n s t e i n p o l y n o m i a l s of type (6) i m p l y on ly the e s t i m a t e O(n-Ce/~). Since the u n i f o r m a p p r o x i m a t i o n by p o l y n o m i a l s Brl(x) of the func t ion x ce fo r a ~ 1 and a ~ 0 cannot have o r d e r o (1 /n ) , T h e o r e m 4 shows tha t the o r d e r of u n i f o r m a p p r o x i m a t i o n of th is func t ion c e a s e s to i m p r o v e a s c~ i n c r e a s e s , and s t a r t i n g not with c~ = 2, bu t s i g n i f i c a n t l y e a r l i e r , v im, when

b e c o m e s l a r g e r than one .

3. In ob t a in ing Eq. (3), we app ly a w e l l - k n o w n m e t h o d in a p p r o x i m a t i o n t h e o r y which c o n s i s t s in r e p l a c - ing the func t ions f(x) u n d e r c o n d i e r a t i o n by t h e i r S tek lov m e a n s . The me thod of p r o o f of T h e o r e m s 2-.4 r e f e r - r i n g to the B e r n s t e i n p o l y n o m i a l s d i f f e r s f r o m tha t u s u a l l y e m p l o y e d in c o n s t r u c t i v e func t ion t h e o r y . I n s t ead of a d i r e c t s tudy of the d i f f e r e n c e b e t w e e n the c o n c r e t e func t ions c o n s i d e r e d and t h e i r ]3e rns t e in p o l y n o m i a l s , which i nvo lves o v e r c o m i n g a n a l y t i c d i f f i cu l t i e s tha t a r i s e and does not p e r m i t us to d i s c o v e r the e s t i m a t e s which hold , we h e r e ob ta in the n e c e s s a r y e s t i m a t e s b y r e d u c i n g the p r o b l e m to the o s c i l l a t i o n s Bk(X) - Bk_l(x ) of a sequer~ce of B e r n s t e i n p o l y n o m i a l s and to the a p p l i c a t i o n of s o m e r a t h e r s i m p l e i n e q u a l i t i e s .

P r o o f of T h e o r e m 1. C o n s i d e r the s econd S tek lov func t ion fhh(X) (cf. [8]). It is obvious that

I E[/(~) ] - - / [ E (~) ] ! = I ~ [/(~) --fi~ (~) 1 - - {/[E(~) ] --/,.,, [E(~) ] } -~E[/h~.(~)]--fi,,[E(~)]]<~-- 2sup I/(x)--],.(x)J~IE[j,,~,(~)]--I,,~[E(~)] ] .

- ~ < z <

But by T a y l o r ' s f o r m u l a

f E [f:,h (~)] - - ]~,h [E (~)]] .~ ~/z sup If~,(x) t . D(~).

Consequer~tly,

! E I I ( ~ ) ] - - f I E ( ~ ) ] I ~ 2 sup i / ( x ) - - f~ ,~ (x ) j -~U. , sup I]~,~(x) l , D ( ~ ) . - - c~ .4 X < cr - - e ~ < : X < ~,

471

M a k i n g u s e now of w e l l - k n o w n e s t i m a t e s [8] f o r If(x) - fhh(X) I and I f~h(X) I we ge t t h a t f o r any h > 0

IE [ / ( ~ ) I - - f lE(~)] I~<0>~(/; h) + ~ . T ~ . o)2n (1; h) �9 D(~).

In o r d e r to o b t a i n (3), i t r e m a i n s o n l y to pu t h = 1/2 ~ ' D - ~ .

p r o o f o f T h e o r e m 2. T h e l e f t i n e q u a l i t y in (8) h o l d s b e c a u s e of a w e l l - k n o w n p r o p e r t y of the p o l y n o m i a l s Bn( f ; x) a c c o r d i n g to w h i c h Bn(f ; x) -< f(x) w h e n e v e r f"(x) -< 0 on [0, 1]. F o r the p r o o f of the r i g h t i n e q u a l i t y of (8), c o n s i d e r the s e q u e n c e of d i f f e r e n c e s B k - l ( x ) - Bk(x ) . I t f o l l o w s a t o n c e f r o m the d e f i n i t i o n of the p o l y - nomiaLs Bn(x) t h a t

B~_, (~) - ~ (x) = ~ , t--~-~ ! - . ~ s \v~-i- Uf - (~3)

F o r f(x) = x l n x -1, we h a v e

, f ~, v [v-1'~ k \ k - 1 l

__~ t ~ l n ~ _ . 7 _ l n v _ A _ _ i lnZ, 1 v - -1 v - : ] = . - - Ink-- ~

-- k (k _---T) ( k - - l ) In - - ( v - - l ) I n v - - _ l j .

T h i s m e a n s t h a t f o r o u r f u n c t i o n h - - !

~' - ( v - 1 ) l ~ 7 = - i j " i v ) x " ( l - x ) ' . . . . . Bh--I (x) --BI~(x) = k(~,_ t) .%~_L v ( k - - 1) lny-L- i

B u t w h e n 1 -< u -< k - 1, we h a v e the i n e q u a l i t y (k - 1) l n k / ( k - 1) <- u l n u / ( u - 1). T h e r e f o r e , f o r s u c h v a l u e s

of u we a l w a y s h a v e

ui (k - - t ) ln k l (k - -1 ) - - (a - - l ) ln u l ( u - - t ) } ~< ( k - - t ) l n k l ( k - - t ) ,

i . e . , t he m a x i m u m of the f u n c t i o n u{(k - 1) l n k / ( k - 1) - (u - 1) l n u / ( u - 1)} on the i n t e r v a l 1 _< u -< k - 1 i s

a t t a i n e d f o r u = 1. H e n c e

', ~ i [k\ ~ ; , - v I - - ~ - - t~ - - ~/< ( 1 4 ) o <~ B~ (.~') - - B.~_, (~) < ~ (~< - - 'U ~, [ , ' } x" ( l - ~, - - ~< o< - u

I nequa l i t y (14) leads to the es t ima te

<' l - ~ - ( ~ - W~ = _ L l l _ ~ (z) _ %,(! _ ~)], z l n - : - - - B , ( x ) < ~ ,, k (k - - 1) n

h=n~, t

w h e r e ~ (x) = nk= - �9 k ( s 1)" I t r e m a i n s to t a k e in to a c c o u n t t h a t

P r o o f of T h e o r e m 3. T h e o r e m 2. t h a t

x x

~ xr~ f tn--[ i ~u n ZL k ( k - - - - - t ) = n ( x - - t ) ~ d t = ( ' l - - x ) ( ~ d t .

In o b t a i n i n g e s t i m a t e (11), we h e r e u s e the s a m e m e t h o d a s t h a t u s e d in p r o v i n g C o n s i d e r t h e s e q u e n c e of d i f f e r e n c e s Bk_ i (x ) - Bk(X). W e h a v e f o r t h e f u n c t i o n f(x) = x a (0 < a < 1)

1 v~r f I k r I ~ ' k ' - ~ ( k - - i T - - - " ' - ~ ( " - - ~ ? " - ~ . ( k ~)------~'~ { ~ t - C - ) - - ~ : ~ , J J '

w h e r e (p(x) = [i - (1 - x ) O q / x .

T h e r e f o r e ,

B , _ ~ ( z ) - - B k ( z ) - (§ ~ 4" '((~ ~ - 6 - ) - - c; x " (5 - - z ) ~ . . . . . k(k - - i) ",.,=1

L e t u s s h o w t h a t w h e n I -< v -< k - 1, t h e i n e q u a l i t y

,-~ {~( t / , , ) -~ (ifk)} <~ 1 -~ ( t t k )

472

i s v a l i d .

o r

I n d e e d , to do th i s i t i s s u f f i c i e n t f o r u s to e s t a b l i s h tha t i f 0 <- t _~ u < 1, t h e n ~2(u) - q~(t) _< uC~[1 - ~ ( t ) ] ,

[ i - ~ (~) ]1[ l - ~ ~ ]~> ~ - ~ (t).

W e o b s e r v e tha t 1 - ~0(t) d e c r e a s e s on [0, 1] and ~p(0) = ~. H e n c e , the i n e q u a l i t y we n e e d w i l l ho ld if we d e m o n s t r a t e tha t ,1 - q)(u) >- (1 - ~ ) ( 1 - u a) f o r 0 - u -< 1, o r ~0(u) -< c~ + (1 - ( ~ ) u a , i . e . , 1 - (1 - u) e2 _<

a u + (1 - a ) u l + a , i ' w h e r e 0 < a < 1. L e t us s h o w tha t in f a c t we h a v e the s t r o n g e r i n e q u a l i t y

l--(t--u)"<~.au-}-('J.--o~)u ~- (0~<u-<.t , 0 < ~ < t ) .

W r i t i n g v = 1 - u, s u c h an i n e q u a l i t y t a k e s the f o r m 1 - v ~ _< ~(1 - v) + (1 - e~)(1 - 2v + v 2) o r , a f t e r s o m e

t r a n s f o r m a t i o n , (2 - ~ ) v l-c~ - (1 - (~)v 2-(~ -< 1 (0 -< v _<_< 1). B u t t h i s i n e q u a l i t y f o l l o w s f r o m t h e f a c t t h a t the d e r i v a t i v e o f t he f u n c t i o n (2 - ~ ) v ~-(~ - (1 - ~ ) v ~-c~ i s e q u a i t o ( 2 - c~) (1- c~ )v -c~ (1 - v ) a n d i s n o n n e g a t i v e on

the i n t e r v a l 0 < v -< 1. T h u s ,

0 <~B~ (x) --B~-~ (x) ~< [ (,V - ~ ' - (~;-- l ) '-~) 1~:] { i - - x " - - (,~--x) ~}.

I t f o l l o w s f r o m th i s t ha t

o ~<B~(x) - ~ _ ~ (z) ~< [ ( . i -~)/~;(~;-~ ) -] { f - ~ ~ - (- l- ,~)3,

w h e r e 0 < oz < 1. T h e l a s t i n e q u a l i t y l e a d s to t he e s t i m a t e

x vc (~

O,~x B, . (x)= ~ IBu(x)--B~_~(x)!<~(1--~) ~ --k(l~--.t) ~- ' h ~ n --I Cirri-=[

(15)

f r o m w h i c h i t f o l l o w s tha t

- ~ z - -B,~(~ ) --O((1--~), , 'n ~) (16)

u n i f o r m l y irl x on [0, 1] and f o r a l l a in t h e i n t e r v a l 0 < a 0 <_ a -< 1. W e now s h o w tha t the e s t i m a t e e s t a b l i s h e d by (16) g i v i n g an u p p e r b o u n d f o r the o r d e r of d e c r e a s e of the q u a n t i t y 0 max [x:--B,(z)] , w h e r e 0 < c~ < 1, i s

~ x ~ < t p r e c i s e . In f a c t , s i n c e

we h a v e

k - - I

But the f u n c t i o n (p(t) i n c r e a s e s on [0, 1] f o r O < ~ < 1. T h e r e f o r e ,

x ~ - B~ (~) = ~ IB~ (~) -- B,,_~ (x)l ~> x ~ (~ - ~ i - - ~ h=n+l t~ .=n

Consequent ly , put t ing Xn = l / n , we have

~ax i x ~ - - ~ (~)]>~ x~ - - B,, (-,~) >~

If we a l s o t a k e a c c o u n t of t he f a c t tha t

lira [l--q)(t/n)] { ( l - - x , , ) ' - - (t--x~,) 2''} = ( l - - a ) (e-'--s-S), r t ~ o O

w e o b t a i n f r o m th i s t h a t

max [x~--B.(x)]~c(t--~z)/n ~ 0 ~ x ~ t

(2n -- t )~

.1 - ~ ( 1 / , ) [ ( 1 - ~r, , )" - - ( t - - z ~ ) 2 " } . ( 2 n - - t ) ct

h o l d s f o r any p o s i t i v e a < 1, w h e r e c > 0 i s s o m e c o n s t a n t not d e p e n d i n g on n and a . T h i s c o n c l u d e s the p r o o f o f T h e o r e m 3. T h e p r o o f o f T h e o r e m 4 p r o c e e d s in t he s a m e way , e x c e p t tha t we u s e t he i n e q u a l i t y v(~0(1/k - (p(1/v)} -< a - 1 (1 _< v --< k - 1, 1 < a -< 2), w h i c h is e s t a b l i s h e d a n a l o g o u s l y .

1 .

2.

L I T E R A T U R E C I T E D

A. F . T i m a n , A p p r o x i m a t i o n T h e o r y of F u n c t i o n s o f a R e a l V a r i a b l e [in R u s s i a n ] , F i z m a t g i z , M o s c o w (1960).

D. D. S t a n c u , " U s e of p r o b a b i l i s t i c m e t h o d s in t he t h e o r y of u n i f o r m a p p r o x i m a t i o n of c o n t i n u o u s f u n c - t i o n s , " k e y . R o u m a i n e Math . P u r e s A p p l . , 1._44, No. 5, 673-691 (1969).

473

3 .

4.

5.

6.

7. 8.

H. Whitney, "On functions with bounded n-th differences," g. Math. Pures Appl., 36, No. 9, 67-95 (1957). H. Berens and G. Lorentz , "Inverse theorems for Bernste in polynomials," Indiana Univ. Math. J., 2_1.1, No. 3, 693-708 (1972). H. Shapiro, "Some Tauber ian theorems with applications to approximation theory, ' Bull. Am. Math. S.c . , 74, No. 3, 500-504 (1968). P. L. Butzer and R. I. Nessel, Four i e r Analysis and Approx}mation, Vol. 1, Academic P re s s , New York (1971). '~ M. Kac, "Une remarque sur les polyn6mes de M. S. Bernstein," Stud. Math., 7, 49-51 (1938). N. I. Akhiezer, Theory of Approximation, Ungar (1956).

A S Y M P T O T I C E X P A N S I O N F O R P R O B A B I L I T Y OF C O N T I N U A T I O N

OF C R I T I C A L B E L L M A N -- H A R R I S P R O C E S S E S

V. A. T o p c h i i UDC 519.21

1. I n t r o d u c t i o n

In this paper branching processes ~(t), t -> 0, in which part icle lifetime depends on their age (Be l lman- Har r i s processes) , will be considered.

Random variables ~(t) assume integer values which we interpret as population size at moment of time t. We descr ibe ~(t) in te rms of development of the population.

Population development begins at the zero moment of time with a single part icle. The remainder con- tinues ei ther by itself or by offspring. Each part icle has a random lifetime ~ with distribution function F(t) = I~{V _< t} (n -> 0and, perhaps, F(0) ~ 0). A particleVs length of life does not depend on the behavior of the other

par t ic les . At its moment of death, each part icle leaves k descendants with probability hi, ( _ h~ = I indepen-

dently of the other par t ic les and of its own age. We denote the corresponding random variable by ~. The possibil i ty of death of a part icle and the generation of new ones at the moment of generation (this corresponds to the ease F(0) e 0) introduces the charac te r i s t i c feature of the definition of ~(t). For such processes , any part icle , with positive probability, has an a rb i t r a r i ly long branch of descendants with zero duration of life. Since par t ic les with zero l ifetimes die at the moment of generation and generate new part icles , it is then natural not to include them in the population count. Therefore , we shall set g(t) equal to the number of par - t i t l es which exist at moment of time t but do not die at this moment of time. We remark that if, for some

to, ~(t 0) = 0 then, for any t -> t 0, ~(t) = 0.

A more detailed descript ion of the given model can be found in [1, 2] (model 3).

The basic goal of this paper is to obtain asymptotic expansions for the probability of continuation of a c r i t ica l (M~ = 1) process ~(t) in the d iscre te case, i .e . , when 77 is integer-valued (this means that ~(t) = ~([t]) and that the p rocess can be considered as ~(n), n = 0, ~). The method used in our paper allows us to give a simple proof of the resul t of Goldstein [3], formulated below (as Theorem 1). The asser t ion of Theorem 1, but with more constrained assumptions, was f i rs t obtained by Sevast 'yanov [4].

THEOREM1. L e t t 2 ( 1 - F ( t ) ) ~ 0 a s t - - - o o , t e t M ~ = l , t e t D ~ < ~ , b u t l e t D ~ ~0 . Then, i f w e u s e t h e

notation p = M~/ and B = ~ k(k-- l ) h~, the following equation holds: k=0

lim tP(~(t) > 0 ) ~-2p/B. (1) t ~

Trans la ted f rom Sibirskii Matematicheskii Zhurnal, Vol. 18, No. 3, pp. 665-674, May-June, 1977. Original ar t ic le submitted June 25, 1975.

This material is protected by copyright registered in the name o f Plenum Publishing Corporation, 227 West 17th Street, New York, N.Y. 10011. No part o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission o f the publisher. A copy o f this article is available from th e publisher for $ Z 50.

474