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Mean convergence theorems for weighted sums of random variables under a condition of weighted integrability
Journal of Inequalities and Applications volume 2013, Article number: 558 (2013)
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.
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 of integrable random variables is said to be uniformly integrable if
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: Cesà ro uniform integrability.
Definition 1.2 A sequence of integrable random variables is said to be Cesà ro uniformly integrable if
In the following, let and be two sequences of integers (not necessary positive or finite) such that for all and as . Let be a sequence of positive numbers such that as .
Ordóñez Cabrera [3] introduced the concept of uniform integrability concerning an array of constant weights.
Definition 1.3 Let be an array of random variables and be an array of constants with for all and some constant . The array is said to be -uniformly integrable if
Ordóñez Cabrera [3] proved that the condition of uniform integrability concerning an array of constant weights is weaker than uniform integrability, and leads to Cesà ro 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 Cesà ro-type uniform integrability with exponent r.
Definition 1.4 Let be an array of random variables and . The array is said to be Cesà ro-type uniformly integrable with exponent r if
Note that the conditions of Cesà ro uniform integrability and Cesà ro-type uniform integrability with exponent r are equivalent when , , and . Sung [4] obtained the weak law of large numbers for an array satisfying Cesà ro-type uniform integrability with exponent r for some .
Chandra and Goswami [5] introduced the concept of Cesà ro α-integrability () and showed that Cesà ro α-integrability, for any , is weaker than Cesà ro uniform integrability.
Definition 1.5 Let . A sequence of random variables is said to be Cesà ro α-integrable if
Under the Cesà ro α-integrability condition for some , Chandra and Goswami [5] obtained the weak law of large numbers for a sequence of pairwise independent random variables. They also proved that Cesà ro α-integrability for appropriate α is also sufficient for the weak law of large numbers to hold for certain special dependent sequences of random variables.
Ordóñez 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 Cesà ro uniform integrability, -uniform integrability and Cesà ro α-integrability.
Definition 1.6 Let be an array of random variables and be an array of constants with for all and some constant . Moreover, let be an increasing sequence of positive constants with as . The array is said to be h-integrable with respect to the array of constants if
Under appropriate conditions on the weights, Ordóñez 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 et al. [7] introduced the notion of h-integrability with exponent r ().
Definition 1.7 Let be an array of random variables and . Moreover, let be an increasing sequence of positive constants with as . The array is said to be h-integrable with exponent r if
Sung et al. [7] proved that h-integrability with exponent r () is weaker than Cesà ro-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 Cesà ro - integrability (, ) and showed that residual Cesà ro -integrability for any is strictly weaker than Cesà ro α-integrability.
Definition 1.8 Let , . A sequence of random variables is said to be residually Cesà ro -integrable if
Under the residual Cesà ro -integrability condition for some appropriate α and p, Chandra and Goswami [8] obtained -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 and be an array of random variables. Moreover, let be an array of constants and an increasing sequence of positive constants with as . The array is said to be residually -integrable with respect to the array of constants if
Remark 1.1
-
(i)
The residual -integrability concerning the arrays of constants was defined by Yuan and Tao [9], who called it the residual h-integrability, and was extended by Ordóñez Cabrera et al. [10] to the conditionally residually h-integrability relative to a sequence of σ-algebras.
-
(ii)
If is h-integrable with exponent r, then it is residually -integrable with respect to the array of constants satisfying , , .
-
(iii)
Residually -integrable with respect to the array of constants is weaker than residually Cesà ro -integrable.
-
(iv)
The concept of residually -integrable concerning the array of constants is strictly weaker than the concept of h-integrable concerning the array of constants and h-integrable with exponent r.
Therefore, the concept of residually -integrable concerning the array of constants is weaker than the concept of all Definitions 1.1-1.7, and leads to residual Cesà ro -integrability as a special case.
For the array of random variables, weak laws of large numbers have been established by many authors (referring to: Sung et al. [7, 11]; Sung [4]; Ordóñez 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 -integrable with respect to the array of constants . Our results extend and sharpen the results of Sung et al. [7], Sung et al. [11], Sung [4], Ordóñez 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
An infinite family of random variables is said to be pairwise NQD if every two random variables and () are NQD. The array is said to be rowwise pairwise NQD if every positive integer n, the sequence of random variables 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 be a sequence of pairwise NQD random variables. Let be a sequence of increasing functions. Then is a sequence of pairwise NQD random variables.
If random variables X and Y are NQD, then , so we have the following.
Lemma 2.2 Let be a sequence of pairwise NQD random variables with and , . Then
Using the above lemma, Chen et al. [14] obtained the following inequality.
Lemma 2.3 Let be a sequence of pairwise NQD random variables with and , , where . Then
where depends only on p.
3 Main results and proofs
Theorem 3.1 Let and be an array of random variables. Let be an array of constants and an increasing sequence of positive constants with as . Assume that the following conditions hold:
-
(i)
is residually -integrable concerning the array ;
-
(ii)
.
Then
in and, hence, in probability as , where if and if , where , , , and , .
Proof If and/or , by the -inequality, Jensen’s inequality and , we have
Therefore, if , we have , so a.s. converges for all . If , by Theorem 2.17 of Hall and Heyde [15], we can get that a.s. converges for all . Thus a.s. converges for all in the case of and/or .
Let and for , .
Case . By the -inequality, we obtain
Noting that for all , , by the -inequality, we obtain
Since and , we have
So, in and hence in . Therefore, the proof is completed when .
Case . Since
But
By Burkholder’s inequality (Theorem 2.10 of Hall and Heyde [15]), we have
hereinafter, C always stands for a positive constant not depending on n which may differ from one place to another, thus in and hence in . Therefore, the proof is completed when .
Case . By Burkholder’s inequality, the -inequality and Jensen’s inequality, we have
But
and
Therefore, the proof is completed when . □
Remark 3.1
-
(i)
Putting , , , if is an array of h-integrability with exponent r (), then it is residually -integrable concerning the array . Thus, Theorem 3.1 and Corollary 3.1 of Sung and Lisawadi and Volodin [7] can be obtained from Theorem 3.1.
-
(ii)
Let , , , , , , similar to that of Remark 1 of Ordóñez 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 and be an array of rowwise pairwise NQD random variables. Let be an array of constants and an increasing sequence of positive constants with as . Assume that the following conditions hold:
-
(i)
is residually -integrable concerning the array .
-
(ii)
.
Then
in and hence in probability as .
Proof The proof is similar to that of Theorem 3.1, we can get a.s. converges for all in the case of and/or . Let and as in Theorem 3.1. Without loss of generality, we can assume that for , , then and are arrays of rowwise NQD random variables by Lemma 2.1. Observe that
By Lemma 2.3 and , we have
then in and hence in .
By Lemma 2.3 and , we have
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 r th uniform integrability for pairwise NQD sequence.
Remark 3.3
-
(i)
Let , , , , 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.
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(iii)
Theorem 1 and Corollary 1 of Ordóñez Cabrera and Volodin [6] can be obtained from Theorem 3.2.
Remark 3.4 Putting , , , if is an array of h-integrability with exponent r (), then is residually -integrable concerning the array . Theorem 3.3 and Corollary 3.3 of Sung et al. [7] can be obtained from Theorem 3.2 since an NA sequence is an NQD sequence.
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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.
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Cao, L. Mean convergence theorems for weighted sums of random variables under a condition of weighted integrability. J Inequal Appl 2013, 558 (2013). https://doi.org/10.1186/1029-242X-2013-558
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DOI: https://doi.org/10.1186/1029-242X-2013-558