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

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R E S E A R C H

Open Access

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 . A sequence{Xn,n≥}of integrable random variables is said to be

uni-formly integrable if

lim

a→∞sup_n_≥_E|Xn|I

|Xn|>a

= .

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

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

Deﬁnition . A sequence{Xn,n≥}of integrable random variables is said to be Cesàro

uniformly integrable if

lim a→∞sup_n_≥_



n

i=

E|Xi|I

|Xi|>a

= .

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In the following, let{un,n≥}and{vn,n≥}be two sequences of integers (not

nec-essary positive or ﬁnite) such thatvn>unfor alln≥ andvn–un→ ∞asn→ ∞. Let

{kn,n≥}be a sequence of positive numbers such thatkn→ ∞asn→ ∞.

Ordóñez Cabrera [] introduced the concept of uniform integrability concerning an ar-ray of constant weights.

Deﬁnition . Let{Xni,un≤i≤vn,n≥}be an array of random variables and{ani,un≤

i≤vn,n≥}be an array of constants with

vn

i=un|ani| ≤Cfor alln∈Nand some constant

C> . The array{Xni,un≤i≤vn,n≥}is said to be{ani}-uniformly integrable if

lim a→∞sup_n_≥_

vn

i=un

|ani|E|Xni|I

|Xni|>a

= .

Ordóñez Cabrera [] proved that the condition of uniform integrability concerning an array of constant weights is weaker than uniform integrability, and leads to Cesàro 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 [] introduced the concept of Cesàro-type uniform integrability with exponentr. Deﬁnition . Let{Xni,un≤i≤vn,n≥}be an array of random variables andr> . The

array{Xni,un≤i≤vn,n≥}is said to be Cesàro-type uniformly integrable with exponent

rif

sup n≥



kn vn

i=un

E|Xni|r<∞ and lim a→∞sup_n_≥_



kn vn

i=un

E|Xni|rI

|Xni|r>a

= .

Note that the conditions of Cesàro uniform integrability and Cesàro-type uniform in-tegrability with exponentrare equivalent whenun= ,vn=n,n≥ andr= . Sung []

obtained the weak law of large numbers for an array{Xni}satisfying Cesàro-type uniform

integrability with exponentrfor some  <r< .

Chandra and Goswami [] introduced the concept of Cesàroα-integrability (α> ) and showed that Cesàroα-integrability, for anyα> , is weaker than Cesàro uniform integra-bility.

Deﬁnition . Letα> . A sequence{Xn,n≥}of random variables is said to be Cesàro α-integrable if

sup n≥ 

n

i=

E|Xi|<∞ and lim n→∞



n

i=

E|Xi|I

|Xi|>iα

= .

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Ordóñez Cabrera and Volodin [] introduced the notion ofh-integrability for an array of random variables concerning an array of constant weights, and proved that this con-cept is weaker than Cesàro uniform integrability,{ani}-uniform integrability and Cesàro α-integrability.

Deﬁnition . Let{Xni,un≤i≤vn,n≥}be an array of random variables and{ani,un≤

i≤vn,n≥}be an array of constants with

vn

i=un|ani| ≤Cfor alln∈Nand some constant

C> . Moreover, let{h(n),n≥} be an increasing sequence of positive constants with

h(n)↑ ∞asn↑ ∞. The array{Xni,un≤i≤vn,n≥}is said to beh-integrable with respect

to the array of constants{ani}if

sup n≥

vn

i=un

|ani|E|Xni|<∞ and lim n→∞

vn

i=un

|ani|E|Xni|I

|Xni|>h(n)

= .

Under appropriate conditions on the weights, Ordóñez Cabrera and Volodin [] proved thath-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.

Sunget al.[] introduced the notion ofh-integrability with exponentr(r> ).

Deﬁnition . Let{Xni,un≤i≤vn,n≥} be an array of random variables andr> .

Moreover, let{h(n),n≥}be an increasing sequence of positive constants withh(n)↑ ∞ asn↑ ∞. The array{Xni}is said to beh-integrable with exponentrif

sup n≥



kn vn

i=un

E|Xni|r<∞ and lim n→∞



kn vn

i=un

E|Xni|rI

|Xni|r>h(n)

= .

Sunget al.[] proved thath-integrability with exponentr(r> ) is weaker than Cesàro-type uniform integrability with exponentr, and obtained weak law of large numbers for an array of dependent random variables (martingale diﬀerence sequence or negatively as-sociated random variables ) satisfying the condition ofh-integrability with exponentr.

Chandra and Goswami [] introduced the concept of residual Cesàro (α,p)- integrability (α> ,p> ) and showed that residual Cesàro (α,p)-integrability for anyα>  is strictly weaker than Cesàroα-integrability.

Deﬁnition . Letα> ,p> . A sequence{Xn,n≥}of random variables is said to be

residually Cesàro (α,p)-integrable if

sup n≥ 

n

i=

E|Xi|p<∞ and lim n→∞



n

i=

E|Xi|–iα

p

I|Xi|>iα

= .

Under the residual Cesàro (α,p)-integrability condition for some appropriateαandp, Chandra and Goswami [] obtainedL-convergence and the weak law of large numbers for a sequence of dependent random variables.

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Deﬁnition . Let r>  and{Xni,un≤i≤vn,n≥}be an array of random variables.

Moreover, let{ani,un≤i≤vn,n≥}be an array of constants and{h(n),n≥}an

increas-ing sequence of positive constants withh(n)↑ ∞asn↑ ∞. The array{Xni,un≤i≤vn,n≥

}is said to be residually (r,h)-integrable with respect to the array of constants{ani}if

sup n≥

vn

i=un

|ani|rE|Xni|r<∞ and lim n→∞

vn

i=un

|ani|rE

|Xni|–h(n)

r

I|Xni|>h(n)

= .

Remark . (i) The residual (,h)-integrability concerning the arrays of constants was deﬁned by Yuan and Tao [], who called it the residualh-integrability, and was extended by Ordóñez Cabreraet al.[] to the conditionally residuallyh-integrability relative to a sequence ofσ-algebras.

(ii) If{Xni,un≤i≤vn,n≥}ish-integrable with exponentr, then it is residually (r,h/r

)-integrable with respect to the array of constants {ani}satisfyingani =kn–/r,un≤i≤vn,

n≥.

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

residually Cesàro (α,p)-integrable.

(iv) The concept of residually (r,h)-integrable concerning the array of constants{ani}is

strictly weaker than the concept ofh-integrable concerning the array of constants{ani}

andh-integrable with exponentr.

Therefore, the concept of residually (r,h)-integrable concerning the array of constants {ani}is weaker than the concept of all Deﬁnitions .-., and leads to residual Cesàro

(α,p)-integrability as a special case.

For the array{Xni,un≤i≤vn,n≥}of random variables, weak laws of large numbers

have been established by many authors (referring to: Sunget al.[, ]; Sung []; Ordóñez Cabrera and Volodin []).

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 Sunget al.[], Sunget al.

[], Sung [], Ordóñez Cabrera and Volodin [].

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 . Two random variablesXandYare said to be negatively quadrant depen-dent (NQD) or lower case negatively dependepen-dent (LCND) if

P(X≤x,Y≤y)≤P(X≤x)P(Y≤y) ∀x,y∈R.

An inﬁnite family of random variables{Xn,n≥}is said to be pairwise NQD if every

two random variablesXi andXj (i =j) are NQD. The array{Xni,i≥,n≥} is said to

be rowwise pairwise NQD if every positive integern, the sequence of random variables {Xni,i≥}is pairwise NQD.

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Lemma . Let{Xn,n≥}be a sequence of pairwise NQD random variables.Let{fn,n≥}

be a sequence of increasing functions.Then{fn(Xn),n≥}is a sequence of pairwise NQD

random variables.

If random variablesXandYare NQD, thenE(XY)≤EXEY, so we have the following. Lemma . Let{Xn,n≥}be a sequence of pairwise NQD random variables with EXn= 

and EX

n<∞,n≥.Then

E

_n

i=

Xi

 ≤

n

i=

EX_i.

Using the above lemma, Chenet al.[] obtained the following inequality.

Lemma . Let{Xn,n≥}be a sequence of pairwise NQD random variables with EXn= 

and E|Xn|p<∞,n≥,where≤p≤.Then

E

n

i=

Xi

p

≤cp n

i=

E|Xi|p, ∀n≥,

where cp> depends only on p.

3 Main results and proofs

Theorem . Let <r< and{Xni,un≤i≤vn,n≥}be an array of random variables.

Let{ani,un≤i≤vn,n≥}be an array of constants and{h(n),n≥}an increasing sequence

of positive constants with h(n)↑ ∞as n↑ ∞.Assume that the following conditions hold: (i) {Xni,un≤i≤vn,n≥}is residually(r,h)-integrable concerning the array{ani};

(ii) h(n)sup_u

n≤i≤vn|ani| →.

Then

vn

i=un

ani(Xni–bni)→

in Lr_and_,_hence_,_{in probability as n}_{→ ∞}_,_{where b}

ni= if <r< and bni=E(Xni|n,i–)

if≤r< ,wheren,i=σ(Xni,un≤j≤i),un≤i≤vn,n≥,andn,un–={∅,},n≥.

Proof Ifun= –∞and/orvn= +∞, by theCr-inequality, Jensen’s inequality and  <r< ,

we have

sup n≥

vn

i=un

|ani|rE|Xni–bni|r≤sup n≥

vn

i=un

|ani|rE|Xni|r<∞.

Therefore, if  <r< , we haveE(vn

i=un|aniXni|)

r_≤vn

i=un|ani|

r_E_|_X

ni|r<∞, so

vn

i=unaniXni

a.s. converges for alln≥. If ≤r< , by Theorem . of Hall and Heyde [], we can get thatvn

i=unani(Xni–bni) a.s. converges for alln≥. Thus

vn

i=unani(Xni–bni) a.s. converges

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LetX_ni =XniI(|Xni| ≤h(n)) –h(n)I(Xni< –h(n)) +h(n)I(Xni>h(n)) andXni =Xni–Xni =

(Xni+h(n))I(Xni< –h(n)) + (Xni–h(n))I(Xni>h(n)) forun≤i≤un,n≥.

Case  <r< . By theCr-inequality, we obtain

E vn

i=un

aniXni

r ≤E vn

i=un

aniXni

r +E vn

i=un

aniXni

r

.

Noting that |X_ni| ≤(|Xni|–h(n))I(|Xni|>h(n)) for all un ≤i≤vn, n≥, by the Cr

-inequality, we obtain

E vn

i=un

aniXni

r ≤ vn

i=un

|ani|rEXni r

≤

vn

i=un

|ani|rE

|Xni|–h(n)

r

I|Xni|>h(n)

→.

Since|X_ni | ≤min{|Xni|,h(n)}and  <r< , we have

E vn

i=un

aniXni

≤

vn

i=un

|ani|EXni

≤C h(n) sup un≤i≤vn

|ani|

–rvn

i=un

|ani|rE|Xni|r→.

So,vn

i=unaniX

ni→ inLand hence inLr. Therefore, the proof is completed when  <

r< .

Caser= . Since

E vn

i=un

ani

Xni–E(Xni|n,i–) ≤E vn

i=un

ani

X_ni –EX_ni|n,i– +E vn

i=un

ani

X_ni–EX_ni|n,i– ≤E vn

i=un

ani

X_ni –EX_ni|n,i–

+ E

vn

i=un

aniXni.

But

E

vn

i=un

aniXni≤ vn

i=un

|ani|E

|Xni|–h(n)

I|Xni|>h(n)

→.

By Burkholder’s inequality (Theorem . of Hall and Heyde []), we have

E vn

i=un

ani

X_ni –EX_ni|n,i–  ≤C vn

i=un

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≤CE

vn

i=un

a_niEX_ni

|ani|

vn

i=un

|ani|E|Xni| →,

hereinafter,Calways stands for a positive constant not depending onnwhich may diﬀer from one place to another, thusvn

i=unani[X

ni–E(Xni|n,i–)]→ inLand hence inL. Therefore, the proof is completed whenr= .

Case  <r< . By Burkholder’s inequality, theCr-inequality and Jensen’s inequality, we

have

E

vn

i=un

ani

Xni–E(Xni|n,i–)

r

≤CE

vn

i=un

a_niXni–E(Xni|n,i–) 

r/

≤CE

vn

i=un

a_niX_ni –EX_ni|n,i– 

r/

+CE

vn

i=un

|ani|rXni–E

X_ni|n,i–r

≤C

_v

n

i=un

a_niEX_ni 

r/ +C

vn

i=un

EaniXni r

.

But

vn

i=un

a_niEX_ni≤ h(n) sup un≤i≤vn

|ani|

–rvn

i=un

|ani|rE|Xni|r→,

and

vn

i=un

EaniXni r

≤

vn

i=un

|ani|rE

|Xni|–h(n)

r

I|Xni|>h(n)

→.

Therefore, the proof is completed when  <r< .

Remark . (i) Puttingani=kn–/r,un≤i≤vn,n≥, if{Xni,un≤i≤vn,n≥}is an array

ofh-integrability with exponentr( <r< ), then it is residually (r,h/r_)-integrable

con-cerning the array{ani}. Thus, Theorem . and Corollary . of Sung and Lisawadi and

Volodin [] can be obtained from Theorem ..

(ii) Letani= /nr,un= ,vn=n,n≥,h(n) =nα,α∈(, /r), similar to that of Remark 

of Ordóñez Cabrera and Volodin [], Theorem . and . and Corollary . of Chandra and Goswami [] can be obtained from Theorem ..

Theorem . Let≤r< and{Xni,un≤i≤vn,n≥}be an array of rowwise pairwise

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

(i) {Xni,un≤i≤vn,n≥}is residually(r,h)-integrable concerning the array{ani}.

(ii) h(n)sup_u_n_≤_i_≤_v_n|ani| →.

Then

vn

i=un

ani(Xni–EXni)→

in Lr_{and hence in probability as n}_{→ ∞.}

Proof The proof is similar to that of Theorem ., we can getvn

i=unani(Xni–EXni) a.s.

converges for alln≥ in the case ofun= –∞and/orvn= +∞. LetXniandXnias in

The-orem .. Without loss of generality, we can assume thatani>  forun≤i≤vn,n≥,

then{aniXni,un≤i≤vn,n≥}and{aniXni,un≤i≤vn,n≥}are arrays of rowwise NQD

random variables by Lemma .. Observe that

vn

i=un

ani(Xni–EXni) = vn

i=un

ani

X_ni –EX_ni+

vn

i=un

ani

X_ni–EX_ni

=:An+Bn.

By Lemma . and|X_ni| ≤min{|Xni|,h(n)}, we have

E(An)≤C vn

i=un

a_niEX_ni–EX_ni 

≤C

vn

i=un

a_niEX_ni

|ani|

–rvn

i=un

|ani|rE|Xni|r→,

thenAn→ inLand hence inLr.

By Lemma . and|X_ni| ≤(|Xni|–h(n))I(|Xni|>h(n)), we have

E|Bn|r≤C vn

i=un

|ani|rE

X_ni–EX_nir

≤C

vn

i=un

|ani|rEXni r

≤C

vn

i=un

|ani|rE

|Xni|–h(n)

r

I|Xni|>h(n)

→.

Thus, the proof is completed.

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Remark .

(i) Letr= ,ani=k–n ,un≤i≤vn,n≥, then Theorem ., Corollary . of Sunget al.

[] and Theorem . of Yuan and Tao [] can be obtained from Theorem .. (ii) Theorem . of Chandra and Goswami [] can be obtained from Theorem .. (iii) Theorem  and Corollary  of Ordóñez Cabrera and Volodin [] can be obtained

from Theorem ..

Remark . Puttingani=kn–/r,un≤i≤vn,n≥, if{Xni,un≤i≤vn,n≥}is an array

ofh-integrability with exponentr( <r< ), then{Xni,un≤i≤vn,n≥} is residually

(r,h/r)-integrable concerning the array{ani}. Theorem . and Corollary . of Sunget al.

[] can be obtained from Theorem . since an NA sequence is an NQD sequence.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

The authors are very grateful to the referees and the editors for their valuable comments and some helpful suggestions 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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