Note on the strong convergence of a weighted sum

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R E S E A R C H

Open Access

Note on the strong convergence of a

weighted sum

Xiaochun Li

*

_{, Yu Miao and Xiaoman Zhang}

*_{Correspondence:}

L.xiaochun@tom.com College of Mathematics and Information Science, Henan Normal University, Xin Xiang, China

Abstract

In this paper, we consider an interesting weighted sums

μ

n:=_(log1_n₎1–p

n

j=2

Xn+2–j

j(logj)p, 0 <p< 1, where{X,Xn,n≥1}is a sequence of independent and identically

distributed random variables withEX= 0, and the equivalence of the almost sure and complete convergence of

μ

nis proved.

Keywords: strong convergence; complete convergence; weighted sums

1 Introduction

Hsu and Robbins [] introduced the concept of complete convergence as follows: a se-quence{Yn,n≥}converges completely to the constantcif

∞

n=

P|Yn–c|>ε

<∞ for allε> .

They also show that the sequence of arithmetic means of i.i.d. random variables converges completely to the expected value if the variance of the summands is ﬁnite. The converse was proved by Erdös [, ]. Furthermore, ifYn→ccompletely, then the Borel-Cantelli lemma trivially implies thatYn

a.s.

−→c, almost sure convergence asn→ ∞. The converse statement is generally, or even ‘typically’, not true.

There are numerous publications in the literature studying the almost sure convergence and complete convergence for the weighted sums of a sequence of random variables. Gut [] provided necessary and suﬃcient conditions for the complete convergence of the Cesáro means of i.i.d. random variables. Liet al.[] obtained some results on complete convergence for weighted sums of independent random variables. Cuzick [] proved a strong law for weighted sums of i.i.d. random variables. Miao and Xu [] established a general result for the weighted sums of stationary sequence.

Throughout this paper, assume that {X,Xn,n≥}is a sequence of independent and identically distributed random variables withEX= . Chow and Lai [] established the following result.

Theorem .[, Theorem ] Let X,X,X, . . .be i.i.d.random variables such thatEX= .

Forα> ,the following statements are equivalent: . Eexp(t|X|α_{) <}_{∞ ∀}_t_{> }_;

. lim_n_→∞(logn)–/α_X

n= a.e.;

. limn→∞(logn)–/αni=cn–iXi= a.e.for some(or equivalently for every)nonvoid

sequence of real numbers(cn,n≥)such thatcn=O(n–β),whereβ=α/(α+ ).

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As an interesting particular case of Theorem ., we consider the following weighted sums:

μn:=  (logn)–p

n

j=

Xn+–j

j(logj)p,  <p< .

The aim of this paper is to prove the following.

Theorem . We haveμn

a.s.

−→if and only ifμn→completely.

Remark . For the casep= , Wu [] discussed the following weighted sums:

νn:= 

log logn

n

j=

Xn+–j

jlogj

and proved the equivalence of the almost sure and complete convergence of the se-quenceνn. On the one hand, because of the limitation ofαin Theorem ., we here only discuss the case ofp> . On the other hand, in order to prove the equivalence of the al-most sure and complete convergence ofμnfor the casep= , we need the exponential integrability ofX, but from Theorem ., this does not hold.

LetSj=

j

k=

Xk

k(logk)p; then we have the following.

Corollary . Ifμn

a.s. −→,then



(logn)–p_max_≤_j_≤_nSj→ completely. 2 Proofs of main results

Let [·] denote the usual integer part of ‘·’ and assume thatn>n= [e]. The constantC

in the proofs below depends only on the distribution of the underlying random variableX

and may denote diﬀerent quantities at diﬀerent appearances.

2.1 Proof of Theorem 1.2

The proof of Theorem . can be derived from Lemma . and Lemma ..

Lemma . The following estimate holds: n

j=[_logn]

log

 +C

logn j(logj)p



<Clogn.

Proof Sinceg(j) =log( +C(_j₍log_logjn₎p)) decreases inj,

n

j=[logn]

log

 +C

logn j(logj)p



≤

n

 logn

log

 +C

logn x(logx)p



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≤logn

n

logn

 

log

 +C



t(logt+log logn)p



dt+O()

≤logn

∞

 

log

 +C



t(logt+log logn)p



dt+O()

≤Clogn,

which yields the desired result.

Lemma . Ifμn

a.s. −→,then

μ()_n :=  (logn)–p

n

j=[logn]

Xn+–j

j(logj)p →

completely and

 (logn)–p

[_log[nlogn]]

j=[logn]+

Xn+–j

j(logj)p →

completely.

Proof From Theorem ., we knowEe|X|–p

<∞, which impliesEe|X|_<_∞_{. Hence by the}

elementary inequality

eηx–  –ηx ≤ηe|x| for allη∈[, ],

we have

EeηX≤ +ηEe|X|=  +ηC.

Notice that ifn>nandn≥j≥[_logn], then _j₍logn_logj₎p < . Hence, by Lemma ., for any

ε> ,

∞

n=n

Pμ()_n >ε≤

∞

n=n

n–ε(logn)(–p)

Eelogn

n j_{=[ }logn]

Xn+–j j(logj)p

=

∞

n=n

n–ε(logn)(–p) n j=[_logn]

Ee

logn j(logj)pXn+–j

≤ ∞

n=n

n–ε(logn)(–p) n j=[_logn]

 +C

logn j(logj)p



≤ ∞

n=n

n–ε(logn)(–p)eClogn

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Similarly, we can obtain∞_n₌_n_P(μ()n < –ε) <∞, then

∞

n=nP(|μ

()

n |>ε) <∞. The ﬁrst statement now follows if we combine the two inequalities. The same technique yields the

second statement.

Lemma . If

μ()_n :=  (logn)–p

[_logn]

j=

Xn+–j

j(logj)p

a.s. −→,

thenμ()n →completely.

Proof For anyε>  we have  =Plim sup

n→∞

μ()_n >ε

≥Plim sup

m→∞

μ()_n₍_m₎ >ε

,

wheren(m) = [mlogm],m∈N. Since

n(m+ ) +  –

logn(m+ ) >n(m)

for m> , the random variables μ()_n₍_m₎, m= , , . . . are independent. Notice also that (logn(m))–p_{< (}_log_m₎–p_{. So by the Borel-Cantelli lemma and the second statement of} Lemma ., we have∞_m₌_n_P(|μ()_n₍_m₎|>ε) <∞, proving the lemma.

2.2 Proof of Corollary 1.1

We easily see that

ES_n=EX

n

j=



(j(logj)p₎ <C<∞.

Now we apply the inequality

Pmax

≤j≤nSj>η

≤PSn>η–

ES

n

(cf.Chow and Teicher [], p.) by takingη= (logn)–pεforε> . Sinceη–ES

n>η

for allnsuﬃciently large,

P

max

≤j≤n

Sj (logn)–p >ε

≤P

 (logn)–p

n

j=

Xn+–j

j(logj)p ≥  ε

.

Therefore,

∞

n=n

P

max

≤j≤n

Sj (logn)–p >ε

≤

∞

n=n

P

μn>

ε

<∞.

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Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript, read and approved the ﬁnal manuscript.

Acknowledgements

This work is supported by HASTIT (No. 2011HASTIT011), NSFC (No. 11001077), NCET (NCET-11-0945), and Plan For Scientiﬁc Innovation Talent of Henan Province (124100510014).

Received: 11 January 2014 Accepted: 21 April 2014 Published:12 May 2014

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4. Gut, A: Complete convergence and Cesáro summation for i.i.d. random variables. Probab. Theory Relat. Fields97, 169-178 (1993)

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10.1186/1029-242X-2014-179

Cite this article as:Li et al.:Note on the strong convergence of a weighted sum.Journal of Inequalities and Applications