d Journal of Inequalities and Applications

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A note on the almost sure central limit theorem for the product of some partial sums

Yang Chen1, Zhongquan Tan2* and Kaiyong Wang1

"Correspondence: tzq728@163.com 2College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, 314001, PR. China

Full list of author information is available at the end of the article

Abstract

Let (Xn) be a sequence of i.i.d., positive, square integrable random variables with E(X1) = / >0, Var(Xi) = a2. Denote by Sni/k = £n=, X -Xk and by y = aff the coecient of variation. Our goal is to show the unbounded, measurable functions g, which satisfy the almost sure central limit theorem, i.e.,

'")-f**™ a.s.,

lim , , „, ,

n^ot log N ^ n \ \(n K 1)nx

where F(-) is the distribution function of the random variable eN and N is a standard

normal random variable.

MSC: Primary 60F15; secondary 60F05

Keywords: almost sure central limit theorem; partial sums; unbounded measurable functions

£ Spri

ringer

1 Introduction

The almost sure central limit theorem (ASCLT) has been first introduced independently by Schatte [1] and Brosamler [2]. Since then, many studies have been done to prove the ASCLT in dierent situations, for example, in the case of function-typed almost sure central limit theorem (FASCLT) (see Berkes etal. [3], Ibragimov and Lifshits [4]). The purpose of this paper is to investigate the FASCLT for the product of some partial sums.

Let (Xn) be a sequence of i.i.d. random variables and define the partial sum Sn = ^n=1 Xk for n > . In a recent paper of Rempala and Wesolowski [ 5], it is showed under the assumption E(X2) < ro and X > that

nil Sk \ ^ d eV2N (1

n\^n ) ' )

where N is a standard normal random variable, ¡x = E(X) and y = a ¡¡i with a2 = var(X). For further results in this field, we refer to Qi [6], Lu and Qi [7] and Rempala and Wesolowski [8].

Recently Gonchigdanzan and Rempala [9] obtained the almost sure limit theorem related to (1) as follows.

©2014 Chen et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the originalworkis properly cited.

Theorem A Let (Xn) be a sequence ofi.i.d., positive random variables with E(Xi) = / > 0 and Var (X^ = a2. Denote by y = a// the coecient of variation . Then, for any real x,

lim-^ f 1 lif^kh) ^ < x) = G(x) a. s., (2)

N^x logN^ n VV n!/n )~)

where G(x) is the distribution function of'e^N, N is a standard normal random variable. Some extensions on the above result can be found in Ye and Wu [10] and the reference therein.

A similar result on the product of partial sums was provided by Miao [11], which stated the following.

Theorem B Let (Xn) be a sequence ofi.i.d., positive, square integrable random variables with E(X1) = / >0 and Var(X1) = a2. Denote by Sn,k = ^n=1 Xi - Xk and y = a// the coe-cient ofvariation. Then

( nLl Sn,k ) ^ d N M

\(n ?) n/n) ^ e , ()

and for any real x,

lim N1 /ffllk^)^<x) = F(x) as (4) N^x logN^ n \V(n?) nxn) ~ ) '

where F(•) is the distribution function of the random variable eN and N is a standard normal random variable.

The purpose of this paper is to investigate the validity of (4) for some class of unbounded measurable functions g.

Throughout this article, (Xn) is a sequence of i.i.d. positive, square integrable random variables with E(Xi) = ¡¡ > and Var (Xi) = a2. We denote by Sn,k = ^ ti X¡ - Xk and by y = a/¡ the coecient of variation. Furthermore, N is the standard normal random variable, $ is the standard normal distribution function, $ is its density function and a ^ b stands for limsupn^TO |an/bn| < ro.

2 Main result

We state our main result as follows.

Theorem Let g(x) be a real-valued, almost everywhere continuous function on R such that |g(ex)^(x) \ < c( + |x|)-a with some c >0 and a >. Then, for any realx,

Nimx¡ogNf -Lxg(x'dFw as. , ®

where F (•) is the distribution function of the random variable eN. Letf (x) =g(ex). By a simple calculation, we can get the following result.

Remark Let f (x) be a real-valued, almost everywhere continuous function on R such that f (x)^(x)|<c( + |x|)-a with some c >and a >. Then ( 5)isequivalent to

lim 1 'N i ft 1 i Sn,k nTco log nJ\ y (n ?) v,

f (x)$(x) dx a.s. ()

Remark Lu et al. [12] proved the function-typed almost sure central limit theorem for a type of random function, which can include U-statistics, Von-Mises statistics, linear processes and some other types of statistics, but their results cannot imply Theorem 1.

3 Auxiliary results

In this section, we state and prove several auxiliary results, which will be useful in the proof of Theorem 1. Let Sn = ^n=1 and U, = Y^k=1 log Tfifc. Observe that for |x| <we have

log( + x)= x + — x2, where U e (?,). Thus

u ^ ± log Si,k

i k=1 (i ?) v

= 1 ^f Si,k 1 1 A Sik 1

= YYTij^xa?) v- / ?) v-

= /Ej=kj<i(Xj - v) \ 1 e± / Si,k

(i ?) a

Y^i t=1 H(i?) V

k=1 Si,k

1 \ ■■ Xk- v 1 Uk_( _

= 7ik=1 a + Y 7ik=1 n(i ?) v

=: 7>Si + Ri. i

By the law of iterated logarithm, we have for k

(i ?) v

= 0( (log log i/i)1/2) a.s.

Therefore,

|Ri| =

Obviously, E|R,| = E

■T-(-

2 \ (i

Y^i k^2\(i?) v

Vik=1V(i?) v

loglog i -1 « 1,0 a.s. ()

1 e± /_

Y^i k=1 H(i ?) v

1 A / Sik V 1 .A 1 1

V,k=1 V(i?) v / V,k=1 i-1 i1/2

Our proof mainly relies on decomposition (7). Properties (8) and (9) will be extensively used in the following parts of this section.

Lemma LetX and Y be random variables. We write F(x) = P(X < x), G(x) = P(X + Y < x). Then

F(x - e)-P( IY | >e) < G(x) < F(x + e)+P( |Y | > e) for every e > 0 and x.

Proof It is Lemma. of Petrov [13]. □

Lemma Let (Xn) be a sequence ofi.i.d. random variables. Let Sn = ^k<nXk, Fs denote the distribution function obtainedfrom F by symmetrization, and choose L > 0 so large that f\x\<Lx2 dFs > 1. Then, for any n > 1, X >,

sup P a < < a + X < AX

a \ Vn /

with some absolute constant A, provided X^fn > L.

Proof It can be obtained from Berkes et al. [3]. □

Lemma Assume that (6) is true for all indicator functions of intervals and for a fixed a.e. continuous function f (x) =f0(x). Then (6) is also true for all a.e. continuous functions f such that f (x)| < |/0(x)|, x e R, and, moreover, the exceptional set of probability 0 can be chosen universally for all suchf.

Proof See Berkes et al. [3]. □

In view of Lemma 3 and Remark 1, in order to prove Theorem 1, it suces to prove ( 6) for the case whenf (x)^(x) = (+ |x|)-a, a >.Thus, inthefollowingpart, weput f (x)^(x) = (+ |x|)-a, a >and

zk = y, 1f (ui),

i=2k+1 2^+1

5k' = E 1f M> < ii^},

i=2k+1

where < p < 2(a ?).

Lemma Under the conditions of Theorem 1, we have P(5k = 5k i.o.) =.

Proof Let f-1 denote an inverse function off in some interval, and let a, p satisfy < p < 2(a ?). It is easy to check that

{5k = Ztt^^Ui^fk/(logk)p) for some k < i < 2k+1)

/( (2 log k + (a - 2/3) log log k)

i/2 x k

V2n (log k)a/2

(log k)3 {+( log k + (a - 2/3) log log k)1/2}a k

(logk)p

Note that the function f is even and strictly increasing for x > x0. We have

f-1(k/(logk)p) > (2logk + (a-2p)loglogk)1/2.

Observing that k < i < 2k+1 implies k > 2 log i, in view of (8) we get

P(5k = 51 i.o.) < P(|Ui | > (2 loglog i + (a - 2p) log log log i - 0(1))1/2 i.o. Si

> (2 log log i + (a - 2/) log log log i - O(l))11/2 i.o. > (2 loglog i + (a -2/) log log log i - 0(1))1/2 i.o.

where in the last step we use the assumption a - 2p > and a version of the Kolmogorov-Erdos-Feller-Petrovski test (see Feller [14], Theorem ). This completes the proof of Lemma 4. □

Let ak = f-1(k/(log k)p) and let Gi and Fi denote, respectively, the distribution function of Ui and 4.Set

/41 / çVi \2

x x2 dFi(x) - yj x xdFi(x^ ,

ni = sup

Si = sup

Gi(x) - $ iF;(x)-$( -

Clearly, Oi < 1, lim;^«, Oi =.

Lemma Under the conditions o/Theorem 1, we have N 2

£*( 2 «

(log N )23'

Proo/ Observe now that the relation

f (x) d(G1(x) - G2(x))

< sup |f (x)| • sup |G1(x)-G2(x)|

is valid for any bounded, measurable functions ^ and distribution functions G1, G2. Let, as previously, ak = f-1(k/(logk)^). Thus, for any k < i < 2k+1, we obtain that

Efmil/m <

—= / f 2(x) dGi (x) (log kY\ J\x\<au

</ f 2(x) + n,

J\x\<au \ai /

«Lj2(X)d$(X) + n' (lOgk)2^'

a J (log k)2ß k2

where in the last step, we have used the fact that a, < 1, lim,^c a, =. Hence, by the Cauchy-Schwarz inequality, we have

E(£)2 « E

E i f2(x)d®(x) + n.

J2f 2mi\f (Ui) <

L \i=2k+1 X " 7 / \i=2k +1 k+1 x/ nk+1

(log k)f> k2

V=2k +1 / V=2k+1

« If 2k f f 2(x) d$(x) +

2 \ J \x\<ak

(log k)2^

(log k)2ß ni

ex /2 k2

■ dx +

\<ak (+ \x\)2« ' (log k)2ß =Lti i ■

Note that

f* ex/2 ft/2 f* *2/8 1 f* x2 /2

z-rr- dx = + / « *e /8 + 7 xex/2 dx «

Jo (+ \x\)2a Jo J*/2 *2a+1Jt/2

(+ \x\)2a and thus by (10) and (11), we have

*2a+1 '

<ak ( + \x\)2* dx « af+1 «f (ak) a^+1 « (logk)ß+(a+1)/2 '

Now we estimate q,. By Lemma 1, we have that for some e >,

ni = sup

Gi(x)-$

< sup I Gi (x) - Fi (x) I + sup

Fi(x)-&( -

P(Ui < x)-P( -— < x

m -+*) < ^ - K - <x

< P(\R:\ > e) + supjP^-— < x + s^ - P^-— < + ei-

The Markov inequality and (9) imply that ( ) £|Ri| 1

< ^« -UTe ■ In addition, Lemma 2 yields

sup| P( 7 <x+e)- i 7 < x)l« e

Setting s = i 1/3, we have

ni « ¡1/? + ¡1/3 + Si-

Using Theorem 1 of Friedman et al. [15], we get

Hence,

EVi „ V^ i1/6 + ei /1 r\

- «Y <TO, (15)

i=1 i=1

which, coupled with (13), (14) and the fact 2 (a +) > p, yields

( ) 2k+1 S£(5k)2 * £ (log k)k+(a+1)/2+£ a^ n

(log N )23'

which completes the proof. □

Lemma Let % = eS+1 /(U;)/{/(U;) < -k/r}, ^ = EÏÏ+11/(U;)/{/(U;) < (¿p}. Under the conditions o/Theorem 1, we have/or l > l0

Proo/ We first show the following result, for any < i < 2 and real x, y,

/ i \1/4

|P(Ui < x, Uj < y)-P(Ui < x)P(Uj < y)| «(^jj . (16)

Letting p = j, the Chebyshev inequality yields

> P1/4 ) < 1P-1/2£|S;I2 = P1/2. ()

Using the Markov inequality and (9), we have

p( |Rj| > p 1/4) «i

= < p 1/4.

p1/4 j1/2p1/4 j1/4i1/4

It follows from Lemma 1, Lemma 2, (17), (18) and the positivity and independence of (Xn) that

P(Ui < x, Uj < y)

= P( Ui < x, + Rj < y

S; -Si

= P[Ui < x, — + + Rj < y

>p(u, <x,/1-P—== <y

- Ply -2p1/4 <^1— -=== < y) - P

-Sj -Si

> p1/4 - P( |Rj| > p1/4)

> p(lUi < x, -1-P< y) - (4A + 0() + )p1/4

= P(Ui < x)P( ^l-P^ < y) - (4A + 0() + )p1/4.

We can obtain an analogous upper estimate for the first probability in (19) by the same way. Thus

P(Ui < x, Uj < y) = P(Ui < x)P^/T-p —== < y^ - 0(4A + 0() + )p1/4,

where |01 < .A similar argument yields

P(Ui < x)P(Uj < y) = P(Ui < x)P^PS-fS < y) - 0 (4A + O() + )p1/4,

where |0 '| < , and ( 16) follows. Letting Gjx, y) denote the joint distribution function of Ui and Uj, in view of (12), (16), we get for l > l0

cov( f (Ui)l\f (Ui) < ) ,f (uJf (Uj) < l

(log k)pJ,JV j r j (log l)p f i f (x)f (y) d(Gij(x, y)-Gi(x)Gj(y))

J W<ak J W^a;

2-(i-k-1)/4

|x|<a^ J |y|<a;

(log k)p (log l)p'

where the last relation follows from the facts that: f is strictly increasing for x > xo, f (a,) = and k < i < 2k+1, 2l < j < 2l+1. Thus

(log i)l

|cov(| «

-(l-k-1)/4

(log k)fi (log l)fi

Lemma Under the conditions of Theorem 1, letting Zk = è,k - , we have

E(Z1 + ••• + Zn )2 = O

^ (log N )2fi-1 Proof By Lemma 6, we have

, N ^œ.

E E(ZkZi)

1<k<l<N l-k> log N

« ôogNJ»N22' 'ogN = "(I).

On the other hand, letting || • || denote the L2 norm, Lemma 5 and the Cauchy-Schwarz inequality imply

E E(ZkZi)

1<k<l<N l-k<40 log N

< E iziiiiziii

1<k<l<N l-k<40 log N

< E I&lläll

1<k<l<N l-k<40 log N

= E E|Hk*||Hk*+;'1

0<j<40 log N k=1

/ N \ 1/2 / N \ 1/2

< El&f El*if 40 log N

k=1 l=1 N 2

(log N)2ß-1

and Lemma 7 is proved.

4 Proof of the main result

We only prove the property in (6), since, in view of Remark 1, it is sucient for the proof of Theorem 1.

Proof of Theorem 1 By Lemma 7 we have

El Z1 + ^ ZN) = O((logN)1-20,

and thus setting Nk = [exp(kk)] with (fi ?) 1 < k <, we get

Ee(iLi-^l < œ,

and therefore

Zi + ••• + N lim -k = a.s.

k—>to Nk

Observe now that for k < i < 2k+1 we have

Ef ^ U < 5^} = Ljx dG'x

fix) d®(~) + / f(x) Jg, M-*

J ^sak \aw 7 |x|<ak \

Put m = f (x) d$(x). Since a, < 1, limi—TO a, = and ak —to as k — to, we have

f (x) d^i x 1 - m

J x^ak \ai

lim sup

k—TO 2k<i<2k+1

and thus, using (12), we get

Ef (UW(U) < (¿^ - m

Thus we have

<n-rp + ok (1).

(log k)p

E5t = m E - + ok (1), |*k | < 1.

i (log k)p i

i=2k+1 i=2k+1

Consequently, using the relation ^,<L 1/i = logL + O() and (15), we conclude

E(5i + ••• + 5N)

log 2N-1

« N^w 7 + oN (1)

= O( (log N)-p) + on () = on (1),

and thus (20) gives

r 51 + ••• + 5^

lim -——— = m a.s.

k—TO log 2Nk-1

By Lemma 4 this implies 51 + ••• + 5Nk

lim -—-— = m a.s. ()

k—TO log 2Nk-1

The relation X < implies limk—TO Nk-1/Nk =, andthus( 21) and the positivity of 5k yield 51 + ••• + 5N

lim -—— = m a.s., ()

N—TO log 2N-1

i.e., (6) holds for the subsequence {2N+1}. Now, for each N > , there exists n, depending on N, such that n+1 < N < 2n+2. Then

& + & + ••• + fn 1f (Ui) log N < & + & + ••• + Hn+2 log 2n+2 ()

log 2n+1 < log N log 2n+1 < log 2n+2 log 2n+1

by the positivity of each term of (fk). Noting that (n +) log 2 ~ log N ~ (n +) log as N —^ c, we get (6) by (22) and (23). □

Competing interests

The authors declare that they have no competing interests. Authors^ contributions

Allauthors read and approved the finalmanuscript. Author details

1 Schoolof Mathematics and Physics, Suzhou University of Science and Technology, Suzhou, 215009, P.R. China. 2College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, 314001, P.R. China.

Acknowledgements

The authors wish to thank the editor and the referees for their very valuable comments by which the quality of the paper has been improved. The authors would also like to thank Professor Zuoxiang Peng for severaldiscussions and suggestions. Research supported by the NationalScience Foundation of China (No. 11326175), the NaturalScience Foundation of Zhejiang Province of China (No. LQ14A010012) and the Research Start-up Foundation of Jiaxing University (No. 70512021).

Received: 4January 2014 Accepted: 4 June 2014 Published: 23 Jun 2014 References

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10.1186/1029-242X-2014-243

Cite this article as: Chen et al.: A note on the almost sure central limit theorem for the product of some partial sums.

Journal of Inequalities and Applications 2014, 2014:243