~ , ~ 4 . ¸ . ,

E L S E V I E R Statistics & Probability Letters 40 (1998) 343-351

STATISTICS& PROBABILITY

LETTERS

On the convergence of generalized moments in almost sure central limit theorem

I l d a r I b r a g i m o v a'l, M i k h a i l L i f s h i t s b,c,*,l

a St_Petersburg Branch of Steklov Mathematical Institute of Russian Academy of Sciences, 191011, Fontanka, 27, St-Petersbur 9, Russia

b Mancomtech Center, 197372, Komendantskii, 22-2-49, St-Petersbur 9, Russia c U.F.R. de Mathkmatiques, Universit~ de Lille I, 59655 Villeneuve d'Ascq Cedex, France

Received March 1998

Abstract

Let {~k} be the normalized sums corresponding to a sequence of i.i.d, variables with zero mean and unit variance. Define random measures

n

k = l

and let G be the normal distribution. We show that for each continuous function h satisfying f h d G < ~ and a mild regularity assumption, one has

l imfhdO,=fhdG a.s. n ~

(~) 1998 Elsevier Science B.V. All rights reserved

AMS classification: primary 60F15; secondary 60F05; 60F17; 60B12

Keywords: Almost sure limit theorems; Moments; Strong invariance principle

O. Quick introduction in the problem

Let {~1, ~2 . . . . . ~k . . . . } be a sequence of i.i.d, random variables in ~l with mean zero and unit variance. Consider the sequence o f normalized sums

k=l

* Corresponding author.

] Research supported by Russian Foundation of Basic Research, DFG and INTAS, grants N. 96-01-00096, N. 96-15-96199 and N. 95-0099.

0167-7152/98/$ - see front matter (~) 1998 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 7 1 5 2 ( 9 8 ) 0 0 1 3 4 - 5

344 L lbragimov, M. Lifshits / Statistics & Probability Letters 40 (1998) 343-351

converging in law to the standard normal distribution G in ~l. Define the random distributions Q, in ~1 by

1 n 1

Q,(sg) = 7 - ~ ~ ~l~(~k), ?(n) = logn, (0.1)

for each Borel set d . It is natural to call Q, the empirical measure associated with the sequence {fin}. The almost sure central limit theorem is a statement of the kind

P{Qn ~ G} = 1. (0.2)

We refer to Brosamler (1988) and Schatte (1988) for the origins. See also Ibragimov (1996) and further references therein. Lacey and Phillip (1990) showed that (0.2) is true under our assumptions about {~j}. It means that almost surely for each bounded continuous function,

f hdQn -- 1 ~ lh(~k)---~ f hdG. (0.3) T(n) k=l

In this note, we investigate the convergence (0.3) for unbounded functions h. It is natural to call such a statement the convergence of generalized moments.

1. Main result and discussion

We state our result as follows.

Theorem 1. Let {~j} be i.i.d, random variables with ECj = 0, and E~} = 1. Let Qn be empirical measures from (0.1) and let G denote the standard normal distribution. Let A, Ho >0 and let f :[A, c¢)~ •+ be a nondecreasing function such that f (x) exp{-H0x 2} is nonincreasing and

A°~f dG<c~. (1.1)

Then for every continuous function h which satisfies

Ih(x)l~<f(lxl), Ix I~>A, (1.2)

we have

P { ~ m / h d Q n = f h d G ) = l . n (1.3)

We hope that Theorem 1 is a final word in the problem which attracted some interest during last years. 1 We tried to improve his In Schatte (1991) it is proved that Theorem 1 holds with f(x)=exp{ax2}, a<~.

result and were able to get in Ibragimov and Lifshits (1998) the result with f (x)=exp{x2/2- cx}, c>0. One should mention that the proof in Ibragimov and Lifshits (1998) is much easier than the proof of stronger result in this work. When the work on the article just mentioned was finished, we were already able to prove Theorem 1, but there was no more place in the text for this longer proof. After its submission we were kindly informed by Yu. Davydov (Lille) and K. Borovkov (Melbourn) about the appearance of Berkes et al. (1998), where the result is obtained, roughly speaking, for f ( x )=x -5 exp{x2/2} which is, indeed, very close to the optimal assumption (1.1) but still a bit more restrictive.

We emphasize that the difficulties which we have to overcome are essentially due to the weak moment assumptions on Cj. If we would assume slightly more, namely, El~kl 2 log2(l~kl + 3)<c~, the sufficiency of

L lbragimov. M. Lifshits I Statistics & Probability Letters 40 (1998) 343-351 345

(1.1) could be obtained much easier, see Theorem 1 in Berkes et al. (1998) and Remark 4.2 in Ibragimov and Lifshits (1998).

One may ask, whether Theorem 1 is true under the main condition (1.1), without further regularity assump- tions of f . The answer is negative - see Example 3 below.

Although essential part of our technique is inherited from out preceding works, Ibragimov (1996), Ibragimov and Lifshits (1998), such as use of super-exponential subsequences, variance estimates and ergodic theorem, some additional tools were implemented in the present exposition - an estimate in central limit theorem due to Friedman et al. (1966) and, especially crucial, a high-level strong invariance principle due to Einmahl and Goodman (1995).

2. Proofs

The main step on the way towards (1.3) is contained in the following Lemma.

Lemma 2. Let ~b:[0, o c ) ~ E+ be a nondecreasing function such that for some H > 0 the function ~b(x) exp{-Hx 2} is nonincreasing and f~'~ qb dG <co. Then we have

P{li?__.supf~)([x[)dQn(x)<~2f~)([xl)dG(x)) =1. (2.1)

Proof of Theorem 1. First, we deduce Theorem 1 from Lemma 2. Fix an arbitrary e>0, and choose B>A so large that

L ~ f(x)dG(x) e. <

For each M > 0 we define truncation of the function h by

hM (x) = sign (h(x) ) min{]h(x)l,M }.

We choose M so large that

M > sup Ih(x)l Ixl <B

and (using integrability of h with respect to G, which follows from (1.1) and (1.2))

f hM d G - f hdG <<.~.

We have

limsup,__.~ f h dQ. - [ hdG <<, limsup / ( h - hM)dQn

+t im:up f h M d Q n - f h mdG + / ( h M - h ) d G . (2.2)

The last term is bounded by e by the choice of M. The second term is zero by Lacey-Phillip Theorem (0.3). In order to estimate the first term, we use Lemma 2. For each H > H0, set

( f ( x ) , x >~B, dpH(X) = Z [ exp{H( x2 _ B2)} f(B), 0 <~x <~B.

346 I. Ibragimov, M. Lifshits I Statistics & Probability Letters 40 (1998) 343-351

Then 1. (ass(x) is nondecreasing; 2. (ass(x) exp{-Hx 2} is nonincreasing; 3. We have f o (a~/dG < oo. Moreover,

lim i @.(Ixl)dG(x)= fx f(l~l)dG(x)~2~. H---+~ i>~B

All assumptions of Lemma 2 are verified. Since for each H we have

](h - hM)(x)[ <~ Ih(x)[l(l~l >B ) --< (ass(Ix[),

it follows from (2.3) and (2.1) that

f ( h - hM)dQn ~ inf lim sup f (ass(Ixl)dan(x) lim sup t/---+ OO J H #l----* o o J

<. 2 f (an(Ixl)dG(x)<<. 4e,

and (2.2) finally yields

limsup,,__>~ S h dQ. - f h dG <~ 5e.

This is another form of our statement (1.3). Therefore, the reduction of Theorem complete. []

(2.3)

1 to Lemma 2 is

Proof of Lemma 2. Let I , = f(a(Ix[)dQn(x). We want to prove that a.s.

lim sup In ~< 2 i ~b(ixl) da(x). (2.4) n----+ OO d

Let a > 1. Consider a rapidly increasing sequence of integers nm ---- [earn]. We prove a version of (2.4) for this sequence, namely, a.s.

lim sup Into ~< 2 i (a( lxl ) dG(x ). (2.5) m----rOtS) , /

Indeed, (2.5) is sufficient for (2.4), since for each n E [nm-l,nm] we have

in ~(nm) O~ rn < <"

and we can derive (2.4) letting ~---+ 1. So we fix ~ and prove (2.5) Let 6 E (0, 2 l- ) be a small number; let M~ = v / ~ log2 k be a truncation level and let

^ Mk (k = (k = sign((k) min{l(kI,Mk}

be the truncated sum (k. We evidently have

I. Ibragimov, M. Lifshits I Statistics & Probability Letters 40 (1998) 343-351 347

Accordingly,

In ~ ~ k = l k = l k4,(l~kl)l~l;~t > ~ :=i. +/.*.

In order to prove (2.5), we show that a.s.

lim,~sup i, , ~< f 4,(Ix[) dG(x)

and

lim,~o~sup I~* <~ f 4,(Ixl) dG(x).

Inequality (2.6) follows by Borel-Cantelli arguments from

limn~sup E~?n ~< f 4,(Ixl) dG(x)

and

(2.6)

(2.7)

(2.8)

c(log 3 n + 1 )(log 2 n) 3/2 (2.9) Varin ~< (log n) 1-26

We start the calculations by proving (2.8). Let Fk be the distribution function of the r.v. [~1. Let Fk be the distribution function of the r.v. ]~k[. Let ~ be the distribution function of the centered normal law Gk approximating the law of Zk in Friedman

et al. (1966), and ~ - ( x ) = ~k(x) - ~k(-x). Let Ak = supx> 0 ]Fk(x) - ~+(x) I be the rate of normal approximation. Integration by parts yields

/0 /o E4,(l(k [) = 4,(x) d[Fk(x)- 1] = 4,(0) + [1-Fk(x)] d4,(x)

~< 4,(o) + [1 - ~ ( x ) + d,l d4,(x)

/o ~< 4,(o) + [1 - ~?(x)] d4,(x) + A~4,(Mk)

f 4,(Ixl) dGk(x) + Ak4,(Mk).

We use the integrability and monotonicity of the function 4,(x) and obtain for each M >0

4,(M) ~< f~'~ 4,(x) exp{-x2/2} dx f ~ exp{-x2/2} dx ~< cM exp{M2/2}.

(Starting from this place, we denote by c different constants whose values are unimportant for our proof.) In particular, for our remainder term we have

4,(Mk ) <~ cMk exp{M2/2} = c(log k)6(log2 k) 1/2. (2.10)

348

Hence,

EI.-

1. lbraoimov, M. Lifshits / Statistics & Probability Letters 40 (1998) 343-351

y(n) k=l

c ~ - (log k)a(log2 k) 1/2. ~< (1 + o(1)) c~(Ixl)dG(x)+?-- ~ x

Using Theorem 1 in Friedman et al. (1966) (see also Egorov (1973)), we have

-~-- < 00 .

k=l

The Kroneker lemma yields

1 + Ak(log k)a(log2 k) 1/2 lim (logn)a(log 2 n)l/2 ~k=l k = 0. n---+~x~

This means that the remainder term does not perturb asymptotic behavior of logarithmic averages; relation (2.8) is proved.

Now we estimate the covariances in order to obtain (2.9). Let us fix a pair of integers, ! ~< k, and consider the covariance of the r.v.'s ~b(l(t[) and ¢(1(,1). We use the representation

~k = rll, k + ( l / k ) l /2~ l ,

where the r.v. ql, k is independent of ~l. In what follows, we use ql, k as an approximation for ~k. Let ~ = ~t,k be the tnmcation of r/t,k at the level Mk. Evidently,

I~k - 0[ <. I~k - ~z.kl =(t/k)~/21~zl.

Since ~b is nondecreasing, we have

~(l~kl) ~< O(101 +(Uk)l/21(zl). (2.11)

Since the function (o(x)exp{-Hx z } is nonincreasing, we have

~b (Ir~l + ( l/k )l/zl(zl) <<. ~b(lr~l)exp{n(lol + ( l/k )l/=l(z])z -Bible}. (2.12)

We write

E~(l~zl)~(l~k I) = E4~(I~, I)~(l~k I )[1{1<-<<k/o,J,} + 1{1< ><k/o,J,}].

For the first term, combine (2.11) and (2.12). For the second one, just bound both ~b(l~tl),~b([&[) by q~(Mk) to obtain

e (Ig, I)O(l&l) -< E¢a(l~tl)¢a(IOl)exp{2HMk(l/Ic) ~/6 + H(I/Ic)V3} + ~(Mk)2p{l~tl >(k/l)~/3}.

Now, assume that

T :=n(2Mk+l)(l/k)l/6< 1. (2.13)

Then the exponential factor in our estimate can be bounded by 1 + 2T. Use independence of ~l and t/, and apply Chebyshev inequality to obtain

E4,(l~zl)~(l&l) ~< E~(I~zI)E~(I,~I)[1 + aT] + c~(Mk)2(l/k) 2/3.

I. Ibragimov, M. Lifshits I Statistics & Probability Letters 40 (1998) 343-351 349

Similarly,

~< Eq~(l~l + (I/k )'/2l¢zl)l¢l¢,l <~(,/z),~) + 4~(Mk )P{l~z[ > (k/l) 1/3 }

I)exp{2Hl~k I(l/k) ' Iffzl + H(I/k)l~zl2}l~l¢,l <~¢k,,'z),~ + 4>(MO(l/k ) 2/3

E~(l~kl)[1 + 2T] + (o(Mk)(I/k) 2/3.

By combining our estimates, we have

E4~(l~zl)4~(l~,~l) ~< E4)(l~zl)E4)(l~kl)+c4)(Mk)2Me(l/k)~/6+c4)(Mk)Z(Uk)2/3

<~ E4)(lg~l)EeP(lgkl) + c4)(Mk )2(Mk + 1)(l/k) ~/6.

This calculation and (2.10) prove that

Cov(~b(l~t[), ~b(l~kl) ~< cc~(Mk )2(Mk + 1 )(Ilk) 1/6 <~ c(log k)2a(log2 k )3/2( l/k )l/6 (2.14)

provided that (2.13) holds. Notice that it is sufficient for (2.13) to have

l ~< ck(log 2 k)-3 := lk. (2.15 )

If l is closer to k, we proceed in an obvious manner to obtain

Cov(~b([~ll), <k([~k 1) ~< E~b([~t[)~b([~k [) ~< th(Mk )2 ~< c(log k)26 log 2 k. (2.16)

Now, by using (2.14) and (2.16), the variance on the left-hand side in (2.9) becomes

Var [ , = 1 1 (logn)2 Z ,.C°v(4'(l~'l)'4'(l~kl))

l<~l,k<~n

~< (logn)2 Z Z c(l°gk)2~(l°g2k)3/2(l/k)l/6+kl Z c(l°gk)2~l°g2kkl l<~k<~n l~l<~lk lk<~l<~k

~< c(log n)a~(log2 n) 3/2 { 1 } (logn)2 Z k-7/6 '~--~ l-5/6+k-I Z -~

1 <~k<~n I <~l<~k lk 41<~k

c(log 2 n) 3/2 ~< (logn)2-:~ Z { k-l +k-' log(k/lk)}

I<~k<~n

c(1 + log 3 n)(log z n) 3/2 ~< (log n) 1-2~

Therefore, inequality (2.9) is established. We proceed with the proof of (2.7). We refer to Proposition 1 of Einmahl and Goodman (1995), p. 553,

which enables us to construct ~k on a common probability space with a Wiener process W(t) and a positive deterministic sequence trk 7 1 so that

P ~ ~k trkW(k) I >(log2k) -1 and I~kl>(251ogzk) 1/2, infinitely often? =0. L )

350 L Ibragimov, M. Lifshits I Statistics & Probability Letters 40 (1998) 343-351

This means that it is sufficient for us to investigate the sums

1 2 , - 1 1 (IW(k)l ) logn ~ ~(9 \T~ -+(l°g2k)-' '

l<~k<~n

From well-known properties of the Wiener process it follows that for k ~ oo, and with probability one

sup W(t) W(k) tE[k,k+l] tl/2 kl/2 <~ t ~ suP[k,k+l] IW(t)[O(k-3/2) 4- t c sup[k,k+l] IW(t)kl/2- W(k)l

= O((log 2 k)l/Zk -1 ) + O((logk)l/Zk -1/2) = o((log 2 k) -1 ).

Hence, for k large enough and all t E [k, k + 1 ] we obtain

(IW(k)l +(log2k)_l) <~ ¢ flW(t)l ) (9 \ k,/2 k, t 7 +2(1°g2 t ) - ' .

Since the function 4~(x)exp{-Hx 2} is nonincreasing, we have

+ 2(log2 t) - l )

{4HIW(t)I 4H ) ('IW(t)l'~ (IW(t)l~ exp - - 4- - - =q~ (1 4-o(1)). (9 Ik ~ } t 1/2 log 2 t (log 2 t) 2 k , ~ )

Moreover, we have equivalence for the weights

1 [k+l dt k ak t

Hence,

1 (lW(k)l~ f*+' (IW(t)l'] dt 1 - '£(9k~, , ] ~ (gk tl/2 J 7(+o(I)) .

We conclude with

; /? 1 , . 4 ( I + o ( I ) ) n+l (lW(t)l'~ dt 1 ¢(ig(~)l)dv, logn a, ¢ ~. t '/2 ) 7 ~ "c~

where U ( Q = exp{-~/2}W(exp{z}) is the Ornstein-Uhlenbeck process and zn = log(n + I). Since U is a stationary ergodic process, we may apply Birkhoff ergodic theorem

lim sup I**n__,~ ~< ~--,~lim -vl f0~ (9(i f(~)l) d~

g(9(I U(0)I) = f ¢(Ixl)da(x).

Relation (2.7) follows and Lemma 2 is proved completely. []

The following example shows that the regularity assumptions in Theorem 1 are crucial as is the basic integrability condition.

I. Ibragimov, M. Lifshits / Statistics & Probability Letters 40 (1998) 343-351 351

1 Set L, = log 2 n. For each n there E x a m p l e 3. Let ~y be an i.i.d. Bernoul l i sequence, i.e. P { ~ j = + 1} = ~.

exists a posi t ive cont inuous funct ion fn which vanishes on the interval t - u , r U/3,~.rl/31j, satisfies the inequal-

ity f f n d G < ~ 2 - " and for all integer q such that qn-I/2E(LI,/3,L~] we have f,(qn-1/2)>~n 2. Finally, set

f = ~ , f , . Then, by the law o f iterated logari thm,

l im sup f fdQ. >1 l im sup f.(~.)/(n log n) = w . n ~ o c , / ?/----~ o<3

References

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Appl. 18, 180-185. Einmahl, U., Goodman, V., 1995. Clustering behavior of finite variance partial sum processes. Probab. Theory Related Fields 102,

547-565. Friedman, N., Katz, M., Koopmans, L.H., 1966. Convergence rates for the central limit theorem. Proc. Nat. Acad. Sci. USA 56, 1062-

1065. Ibragimov, I.A., 1996. On almost-everywhere variants of limit theorems. Dokl. Math. 54, 703-705. Ibragimov, I.A., Lifshits, M.A., 1998. On the almost sure limit theorems. Theor. Probab. Appl., submitted. Lacey, M., Philipp, W., 1990. A note on the almost sure central limit theorem. Statist. Probab. Lett. 9, 201-205. Schatte, P., 1988. On strong versions of the almost sure central limit theorem. Math. Naehr. 137, 249-256. Schatte, P., 1991. On the central limit theorem with almost sure convergence. Probab. Math. Statist. 11,237-246.