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Mean convergence theorems for weighted sums of random variables under a condition of weighted integrability

Lingqiu Cao*

Correspondence: clq0621@sina.cn Common Required Course Department, Hengyang Financial and Industry Polytechnic, Hengyang,421002, P.R.China

Abstract

In this paper, a new concept of weighted integrability is introduced for an array of random variables concerning an array of constants, which is weaker than other previous related notions of integrability. Mean convergence theorems for weighted sums of an array of dependent random variables satisfying this condition of integrability are obtained. Our results extend and sharpen the known results in the literature. MSC: 60F15

Keywords: mean convergence; weak laws of large numbers; uniform integrability; integrability concerning the weights; weighted sums; negatively quadrant dependent random variables; negatively associated random variables

1 Introduction

The notion of uniform integrability plays the central role in establishing weak laws of large numbers. In this paper, we introduce a new notion of weighted integrability and prove some weak laws of large numbers under this condition.

Definition 1.1 A sequence {Xn, n > 1} of integrable random variables is said to be uniformly integrable if

lim supE\Xn\l(\Xn\ >a) =0.

Landers and Rogge [1] proved the weak law of large numbers under the sequence of pairwise independent uniformly integrable random variables.

Chandra [2] obtained the weak law of large numbers under a new condition which is weaker than uniform integrability: Cesaro uniform integrability.

Definition 1.2 A sequence {Xn, n > 1} of integrable random variables is said to be Cesaro uniformly integrable if

ft Spri

lim sup - VE\Xi\l{\X¡\ > a) = 0.

I—>■ OO 1 M ' J

a^^ n>i n .

ringer

©2013 Cao; 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 original work is properly cited.

In the following, let {un, n > 1} and {vn, n > 1} be two sequences of integers (not necessary positive or finite) such that vn > un for all n > 1 and vn - un ^to as n ^to. Let {kn, n > 1} be a sequence of positive numbers such that kn ^to as n ^ to.

Ordóñez Cabrera [3] introduced the concept of uniform integrability concerning an array of constant weights.

Definition 1.3 Let {Xni, un < i < vn, n > 1} be an array of random variables and {ani, un < i < vn, n > 1} be an array of constants with ^vnUn \ani \< C for all n e N and some constant C >0. The array {Xni, un < i < vn, n > 1} is said to be {ani}-uniformly integrable if

lim sup V\ani\E\Xni\l(\XM \ > a) =0.

E—OO _ i -' V '

a^TO n>1 i=u

Ordóñez Cabrera [3] proved that the condition of uniform integrability concerning an array of constant weights is weaker than uniform integrability, and leads to Cesaro uniform integrability as a special case. Under the condition of uniform integrability concerning the weights, he obtained the weak law of large numbers for weighted sums of pairwise independent random variables.

Sung [4] introduced the concept of Cesaro-type uniform integrability with exponent r.

Definition 1.4 Let {Xni, un < i < vn, n > 1} be an array of random variables and r >0. The array {Xni, un < i < vn, n > 1} is said to be Cesaro-type uniformly integrable with exponent r if

1 vn 1 vn

sup — y E\Xni\r < to and lim sup — E\Xni\rl(\Xni\r > a) = 0.

n>1 kn^1 a n>1 kn ^

i—un i—un

Note that the conditions of Cesaro uniform integrability and Cesaro-type uniform integrability with exponent r are equivalent when un — 1, vn — n, n > 1 and r — 1. Sung [4] obtained the weak law of large numbers for an array {Xni} satisfying Cesaro-type uniform integrability with exponent r for some 0 < r <2.

Chandra and Goswami [5] introduced the concept of Cesaro a-integrability (a > 0) and showed that Cesaro a-integrability, for any a > 0, is weaker than Cesaro uniform integra-bility.

Definition 1.5 Let a > 0. A sequence {Xn, n > 1} of random variables is said to be Cesaro

a-integrable if

1 n 1 n

sup - VE\Xi\ < to and lim - ^EX^IÍ\Xi\ > ia) — 0.

n>l n n >to n ^ '

i—1 i—1

Under the Cesaro a-integrability condition for some a >2, Chandra and Goswami [5] obtained the weak law of large numbers for a sequence of pairwise independent random variables. They also proved that Cesaro a-integrability for appropriate a is also sufficient for the weak law of large numbers to hold for certain special dependent sequences of random variables.

Ordonez Cabrera and Volodin [6] introduced the notion of h-integrability for an array of random variables concerning an array of constant weights, and proved that this concept is weaker than Cesaro uniform integrability, {ani}-uniform integrability and Cesaro a-integrability.

Definition 1.6 Let {Xni, un < i < vn, n > 1} be an array of random variables and {a„i, un < i < vn, n > 1} be an array of constants with ^V—un \ani \< C for all n e N and some constant C > 0. Moreover, let {h(n), n > 1} be an increasing sequence of positive constants with h(n) t to as n tTO. The array {Xni, un < i < vn, n > 1} is said to be h-integrable with respect to the array of constants {ani} if

supV\ani\E\Xni \ < TO and lim \\\ani\E\Xni\l{\Xni \ > h(n)) — 0.

n>1 i_u n^TO i=u

Under appropriate conditions on the weights, Ordonez Cabrera and Volodin [6] proved that h-integrability concerning the weights is sufficient for the weak law of large numbers to hold for weighted sums of an array of random variables, when these random variables are subject to some special kind of rowwise dependence.

Sung etal. [7] introduced the notion of h-integrability with exponent r (r > 0).

Definition 1.7 Let {Xni, un < i < vn, n > 1} be an array of random variables and r > 0. Moreover, let {h(n), n > 1} be an increasing sequence of positive constants with h(n) t to as n t to. The array {Xni} is said to be h-integrable with exponent r if

1 Vn 1 Vn

sup -J2 E\Xni\r < TO and lim — V E\Xni\rI(\Xni\r > h(n)) = 0.

n>1 kn ._ n ^TO kn ._

i—un i—un

Sung etal. [7] proved that h-integrability with exponent r (r >0) is weaker than Cesaro-type uniform integrability with exponent r, and obtained weak law of large numbers for an array of dependent random variables (martingale difference sequence or negatively associated random variables) satisfying the condition of h-integrability with exponent r.

Chandra and Goswami [8] introduced the concept of residual Cesaro (a, p)-integrability (a > 0,p >0) and showed that residual Cesaro (a,p)-integrability for any a > 0 is strictly weaker than Cesaro a-integrability.

Definition 1.8 Let a > 0,p > 0. A sequence {Xn, n > 1} of random variables is said to be residually Cesaro (a,p)-integrable if

1 n 1 n

sup - VE\Xi\p < to and lim - Ve(\Xi\- f )pl(\Xi\> ia) — 0.

n>l n ^ n >to n ^ v ' v '

i—1 i—1

Under the residual Cesaro (a,p)-integrability condition for some appropriate a and p, Chandra and Goswami [8] obtained L1 -convergence and the weak law of large numbers for a sequence of dependent random variables.

We now introduce a new concept of integrability.

Definition 1.9 Let r > 0 and {Xni, un < i < vn, n > 1} be an array of random variables. Moreover, let {ani, un < i < vn, n > 1} be an array of constants and {h(n), n > 1} an increasing sequence of positive constants with h(n) fro as n fro. The array {Xni, un < i < vn, n > 1} is said to be residually (r, h)-integrable with respect to the array of constants {ani} if

sup£\ani\rE\Xni\r < ro and lim \ani\rE{\Xni| - h(nj)7(\Xni| > h(n)) — 0.

n>1 ~ n^ro

i—Un i—Un

Remark 1.1 (i) The residual (1, h)-integrability concerning the arrays of constants was defined by Yuan and Tao [9], who called it the residual h-integrability, and was extended by Ordonez Cabrera et al. [10] to the conditionally residually h-integrability relative to a sequence of a-algebras.

(ii) If {Xni, un < i < vn, n > 1} is h-integrable with exponent r, then it is residually (r, h1/r)-integrable with respect to the array of constants {ani} satisfying ani — k-1/r, un < i < vn, n > 1.

(iii) Residually (r, h)-integrable with respect to the array of constants {ani} is weaker than residually Cesaro (a,^)-integrable.

(iv) The concept of residually (r, h)-integrable concerning the array of constants {ani} is strictly weaker than the concept of h-integrable concerning the array of constants {ani} and h-integrable with exponent r.

Therefore, the concept of residually (r, h)-integrable concerning the array of constants {ani} is weaker than the concept of all Definitions 1.1-1.7, and leads to residual Cesaro (a,^)-integrability as a special case.

For the array {Xni, un < i < vn, n > 1} of random variables, weak laws of large numbers have been established by many authors (referring to: Sung etal. [7,11]; Sung [4]; Ordonez Cabrera and Volodin [6]).

In this paper, we obtain weak laws of large numbers for the array of dependent random variables satisfying the condition of residually (r, h)-integrable with respect to the array of constants {ani}. Our results extend and sharpen the results of Sung et al. [7], Sung et al. [11], Sung [4], Ordonez Cabrera and Volodin [6].

2 Preliminary lemmas

In order to consider the mean convergence for an array of random variables satisfying dependent conditions, we need the following definition.

Definition 2.1 Two random variables X and Y are said to be negatively quadrant dependent (NQD) or lower case negatively dependent (LCND) if

P(X < x, Y < y) < P(X < x)P(Y < y) Vx,y e R.

An infinite family of random variables {Xn, n > 1} is said to be pairwise NQD if every two random variables Xi and Xj (i — j) are NQD. The array {Xni, i > 1, n > 1} is said to be rowwise pairwise NQD if every positive integer n, the sequence of random variables {Xni, i > 1} is pairwise NQD.

This definition was introduced by Alam and Saxena [12] and carefully studied by Joag-Dev and Proschan [13].

Lemma 2.1 Let {Xn, n > 1} be a sequence ofpairwise NQD random variables. Let {fn, n > 1} be a sequence of increasing functions. Then {fn(Xn), n > 1} is a sequence ofpairwise NQD random variables.

If random variables X and Y are NQD, then E(XY) < EXEY, so we have the following.

Lemma 2.2 Let {Xn, n > 1} be a sequence ofpairwise NQD random variables with EXn = 0 and EXn < to, n > 1. Then

e(t,x) <£EXn.

V i=1 / i=1

Using the above lemma, Chen et al. [14] obtained the following inequality.

Lemma 2.3 Let {Xn, n > 1} be a sequence ofpairwise NQD random variables with EXn = 0 and E| Xn |p < to, n > 1, where 1 < p < n. Then

< c^E\Xi\p, Vn > 2,

where cp > 0 depends only on p. 3 Main results and proofs

Theorem 3.1 Let 0 < r <2 and {Xni, un < i < vn, n > 1} be an array of random variables. Let {a„i, un < i < vn, n > 1} be an array of constants and {h(n), n > 1} an increasing sequence of positive constants with h(n) fro as n fro. Assume that the following conditions hold:

(i) {Xni, un < i < vn, n > 1} is residually (r, h)-integrable concerning the array {ani};

(ii) h(n) supun<i<vn \ani \^0. Then

^2ani(Xni - bni) ^ 0

in Lr and, hence, in probability asn ^to, where bni = 0 if 0 < r <1 and bni = E(Xni | Sn,i_i) if 1 < r < n, where %n>i = a(Xni, un < j < i), un < i < vn, n > 1, and Sn,un_1 = {0, n > 1.

Proof If un = -to and/or vn = +to, by the Cr-inequality, Jensen's inequality and 0 < r < n, we have

sup y \ani\rE\Xni - bni\r < 2 su^y \ani\rE\Xni\r < to.

n>1 ._ n>1 ._

i—un i—un

Therefore,if 0 < r < 1, wehaveE(£vjlun \aniXni\)r < J2VnUn \am\rE\Xni\r < to, so £^ aniXni a.s. converges for all n > 1. If 1 < r <2, by Theorem 2.17 of Hall and Heyde [15], we can get that Y^,vu ani(Xni - bni) a.s. converges for all n > 1. Thus ^v—u„ ani(Xni - bni) a.s. converges for all n > 1 in the case of un — -to and/or vn — +ro.

Let X'ni — XniI(\Xni \ < h(n)) - h(n)I(Xni < -h(n)) + h(n)I(Xni > h(n)) and X^ — Xni - Xni — (Xni + h(n))I(Xni < —h(n)) + (Xni - h(n))I(Xni > h(n)) for un < i < un, n > 1. Case 0 < r < 1. By the Cr-inequality, we obtain

Noting that \Xni \ < (\Xni\ - h(n))I(\Xni \ > h(n)) for all un < i < Vn, n > 1, by the Cr-inequality, we obtain

<Y.\a™\rE\Ki\r

\ani\rE(\Xni\ - h(n))rI(\Xni\ > h(n)) ^ 0. Since \Xni\ < min{\Xni\,h(n)} and 0 < r < 1, we have

<i2\am\E\X'n i \

1-r Vn

< chn) sup \ani\) ^\ani\rE\Xni\r ^ 0.

i/__J i'____'

un<i<Vn

So, ^Vnun aniX'ni ^ 0 in L1 and hence in Lr. Therefore, the proof is completed when 0 < r <1. Case r — 1. Since

^^ ani [Xni - E(Xni \^n,i-1)]

< e£aniXni -E(xIi\»n,i-1)]

< E^ani[X'ni -E(Xni\»n,i-1)]

+ ej2anX -^Xni\»n,i-1)]

Vn \ \

+2ej2 \amXni \.

E^\an/Xni\ < X>»\E(\Xni\ - h(n))I(\Xni\ > h(n)) ^ 0.

i—un i—un

By Burkholder's inequality (Theorem 2.10 of Hall and Heyde [15]), we have

Ej2am[X'ni -E{Xni\»n,i-0]

< C^alEX i - EX i\»n,i-0]2

< CE^TalE\X'„i|2

< c(h(n) sup \ani\E\X„i\^ 0,

\ r/.. <i<i;.. '

hereinafter, C always stands for a positive constant not depending on n which may differ from one place to another, thus ^v==Un ani[X'ni - E(X'ni\^n,i_i)] ^ 0 in L2 and hence in L1. Therefore, the proof is completed when r = 1.

Case 1 < r < 2. By Burkholder's inequality, the Cr-inequality and Jensen's inequality, we have

y] ani \_Xni - E(Xni Pn^-O]

J2a2ni[Xni -E(XniPn,i-i)]2

T,a2ni[X'ni -E(X'ni\»n,i-^]2

+ CE^\ani\r\[X'^i -^xn'i\»n,i-0]\r

(Vn 1 r/2 Vn

J2a2niE\X'„i\2 + ^^E\aniXni\r.

Vn 2-r Vn

^aniE\xni\2 < (h(n) sup \ani\)~J2 \am\rE\Xni\r ^ 0,

^ Un<i<Vn '

J2E\aniKi\r <J2\ani\rE(Xni\-h(n))rl(\Xni\>h(n)) ^ 0.

i=Un i=Un

Therefore, the proof is completed when 1 < r <2. □

Remark 3.1 (i) Putting ani = k-1/r, un < i < Vn, n > 1, if {Xni, un < i < Vn, n > 1} is an array of h-integrability with exponent r (0 < r < 2), then it is residually (r, h1/r)-integrable concerning the array {ani}. Thus, Theorem 3.1 and Corollary 3.1 of Sung and Lisawadi and Volodin [7] can be obtained from Theorem 3.1.

(ii) Let a„i = 1/nr, un = 1, Vn = n, n > 1, h(n) = na, a e (0,1/r), similar to that of Remark 1 of Ordonez Cabrera and Volodin [6], Theorem 3.1 and 3.2 and Corollary 3.1 of Chandra and Goswami [8] can be obtained from Theorem 3.1.

Theorem 3.2 Let 1 < r <2 and {Xni, un < i < Vn, n > 1} be an array of rowwise pairwise NQD random Variables. Let {ani, un < i < Vn, n > 1} be an array of constants and {h(n), n >

1} an increasing sequence of positive constants with h(n) as n fro. Assume that the following conditions hold:

(i) {Xni, un < i < vn, n > 1} is residually (r, h)-integrable concerning the array {ani}.

(ii) h(n) supun<i<vn |ani Then

^^ ani(Xni - EXni) ^ 0

in Lr and hence in probability asn ^<x>.

Proof The proof is similar to that of Theorem 3.1, we can get ^V=Un ani(Xni - EXni) a.s. converges for all n > 1 in the case of un = -ro and/or vn = Let X!ni and X^ as in Theorem 3.1. Without loss of generality, we can assume that ani > 0 for un < i < vn, n > 1, then {aniX'ni, un < i < vn, n > 1} and {aniX')[i, un < i < vn, n > 1} are arrays of rowwise NQD random variables by Lemma 2.1. Observe that

Vn Vn Vn

- EXni) = ^^ ani{X'ni - EXn0 + ^^ an

i=Un i=Un i=Un

=: An + Bn.

By Lemma 2.3 and iX'nii < min{|Xnih(n)}, we have

E(An)2 < cY^alEXi -EX'n¡)2

< Cj^alEK)2

2-r Vn

< c(h(n) sup ianii)~J2 ianiirEiXni|r ^ 0,

Un<i<Vn i = Un

then An ^ 0 in L2 and hence in Lr. By Lemma 2.3 and iXn i < (iXnii - h(n))l(iXnii > h(n)), we have

EiBnir < c£ ianiirE{X>i -EX'm)r

< c£ianiirEX'hi\r

< CJ2 ianiirE(iXnii - h(n))rl(iXnii > h(n)) ^ 0.

Thus, the proof is completed. □

Remark 3.2 Theorem 3.2 extended the result in Chen [16] who first obtained the r-the moment convergence under the rth uniform integrability for pairwise NQD sequence.

Remark 3.3

(i) Let r = 1, ani = k-1, un < i < Vn, n > 1, then Theorem 3.2, Corollary 3.2 of Sung et al. [7] and Theorem 2.2 of Yuan and Tao [9] can be obtained from Theorem 3.2.

(ii) Theorem 2.2 of Chandra and Goswami [8] can be obtained from Theorem 3.2.

(iii) Theorem 1 and Corollary 1 of Ordonez Cabrera and Volodin [6] can be obtained from Theorem 3.2.

Remark 3.4 Putting a„i = k-1/r, un < i < Vn, n > 1, if {Xni, un < i < Vn, n > 1} is an array of h-integrability with exponent r (0 < r < 2), then {Xni, un < i < Vn, n > 1} is residually (r, h1/r)-integrable concerning the array {ani}. Theorem 3.3 and Corollary 3.3 of Sung etal. [7] can be obtained from Theorem 3.2 since an NA sequence is an NQD sequence.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

The authors are very gratefulto the referees and the editors for their valuable comments and some helpfulsuggestions

that improved the clarity and readability of the paper.

Received: 4 May 2013 Accepted: 21 October 2013 Published: 25 Nov 2013

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10.1186/1029-242X-2013-558

Cite this article as: Cao: Mean convergence theorems for weighted sums of random variables under a condition of weighted integrability. Journal of Inequalities and Applications 2013, 2013:558