Journal of Applied MathematicsVolume 2012 (2012), Article ID 735973, 10 pagesdoi:10.1155/2012/735973

Research Article

Limiting Behavior of the Maximum of the Partial Sum for Linearly Negative Quadrant Dependent Random Variables under Residual Cesàro Alpha-Integrability Assumption

Jiangfeng Wang and Qunying Wu

College of Science, Guilin University of Technology, Guilin 541004, China

Received 13 September 2011; Accepted 29 December 2011

Academic Editor: Xianhua Tang

Copyright © 2012 Jiangfeng Wang and Qunying Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Linearly negative quadrant dependence is a special dependence structure. By relating such conditions to residual Cesàro alpha-integrability assumption, as well as to strongly residual Cesàro alpha-integrability assumption, some -convergence and complete convergence results of the maximum of the partial sum are derived, respectively.

1. Introduction

The classical notion of uniform integrability of a sequence of integrable random variables is defined through the condition . Landers and Rogge [1] proved that the uniform integrability condition is sufficient in order that a sequence of pairwise independent random variables verifies the weak law of large numbers (WLLNs). Chandra [2] weakened the assumption of uniform integrability to Cesáro uniform integrability (CUI) and obtained -convergence for pairwise independent random variables.

Chandra and Goswami [3] improved the above-mentioned result of Landers and Rogge [1]. They showed that for a sequence of pairwise independent random variables, CUI is sufficient for the WLLN to hold and strong Cesáro uniform integrability (SCUI) is sufficient for the strong law of large numbers (SLLNs) to hold. Landers and Rogge [4] obtained a slight improvement over the results of Chandra [2] and Chandra and Goswami [3] for the case of nonnegative random variables. They showed that, in this case, the condition of pairwise independence can be replaced by the weaker assumption of pairwise nonpositive correlation.

Chandra and Goswami [5] introduced a new set of conditions called Cesáro -integrability (CI()) and strong Cesáro -integrability (SCI()) for a sequence of random variables, which are strictly weaker than CUI and SCUI, respectively. They showed that, for , CI() is sufficient for the WLLN to hold and SCI() is sufficient for the SLLN to hold for a sequence of pairwise independent random variables, which are improvements over the results of Landers and Rogge [4] and the earlier results.

Chandra and Goswami [6] relaxed the condition of CI() to residual Cesáro alpha-integrability (RCI(), see Definition 2.1 below) and the condition of SCI() to strong residual Cesáro alpha-integrability (SRCI(), see Definition 2.3 below) and significantly improved the results of Chandra and Goswami [5].

Recently, Yuan and Wu [7] discussed some limiting behaviors of the maximum of partial sum for asymptotically negatively associated random variables when such random variables are subject to RCI() and SRCI().

In this paper, we will derive some -convergence and complete convergence of the maximum of partial sum for linearly negative quadrant dependent random variables when such random variables are subject to RCI() and SRCI(). These results generalize previous work in the literature.

2. Preliminaries

First let us specify the two special kinds of uniform integrability we are dealing with in the subsequent sections, which were introduced by Chandra and Goswami [6].

Definition 2.1. For , a sequence of random variables is said to be residual Cesáro alpha-integrable (RCI(), in short) if

Clearly, is RCI() for any if is identically distributed with , and is RCI() for any if is stochastically dominated by a nonnegative random variable with for some .

Definition 2.2. For , a sequence of random variables is said to be strongly residual Cesáro alpha-integrable (SRCI(), in short) if

We point out that, is SRCI() for any , provided that is stochastically dominated by a nonnegative random variable with for some and .

The condition of SRCI() is a “strong” version of the condition of RCI(). Moreover, for any , RCI() is strictly weaker than CI(), thereby weaker than CUI, while SRCI() is strictly weaker than SCI(), thereby much weaker than SCUI.

Next, we turn our attention to the dependence structure for random variables. For our purpose, we have to mention a special kind of dependence, namely, negative quadrant dependence.

Definition 2.3 (cf. Lehmann [8]). Two random variables and are said to be negative quadrant dependent (NQD, in short) if for any , A sequence of random variables is said to be pairwise NQD if and are NQD for all and .

Definition 2.4 (cf. Newman [9]). A sequence of random variables is said to be linearly negative quadrant dependent (LNQD, in short) if for any disjoint subsets and positive ,

Remark 2.5. It is easily seen that if is a sequence of LNQD random variables, then is still a sequence of LNQD random variables, where and are real numbers.

The concept of LNQD sequence was introduced by Newman [9]. Some applications for LNQD sequence have been found; see, for example, the work by Newman [9] who established the central limit theorem for a strictly stationary LNQD process. Wang and Zhang [10] provided uniform rates of convergence in the central limit theorem for LNQD sequence. Ko et al. [11] obtained the Hoeffding-type inequality for LNQD sequence. Ko et al. [12] studied the strong convergence for weighted sums of LNQD arrays. Fu and Wu [13] studied the almost sure central limit theorem for LNQD sequences, and so forth. We note that “" means “.”

Lemma 2.6 (cf. Lehmann [8]). Let random variables and be NQD. Then (1); (2); (3)If and are both nondecreasing (or both nonincreasing) functions, then and are NQD.

Lemma 2.7 (cf. Hu et al. [14]). Let be a LNQD sequence of random variables with . Assume that there exists a satisfying for every . Then, there exists a positive constant such that where is a positive constant depending only on .

It is easily seen that when , the above equation still holds true.

Lemma 2.8. Let be LNQD random variables sequences with mean zero. Then for , there exists a positive constant such that

This lemma is easily proved by the results of Zhang [15] and Yuan and Wu [7]. Here we omit the details of the proof.

Lemma 2.9. Let be a centered LNQD random field. Then for any , there exists a positive constant such that for all .

This lemma is due to Zhang [15, Lemma 3.3].

Finally, we give a lemma which supplies us with the analytical part in the proofs of theorems in the subsequent sections.

Lemma 2.10 (cf. Landers and Rogge [4]). For sequences and of nonnegative real numbers, if then for every .

3. Residual Cesáro Alpha-Integrability and -Convergence of the Maximum of the Partial Sum

Let , and let be a strictly positive function defined on (). In this section, we discuss -convergence of the form of for a LNQD sequence of random variables, provided that is RCI() for an appropriate condition.

Our first result is dealing with the case .

Theorem 3.1. Let , and let be a LNQD sequence of random variables. If is RCI for some , then

Proof of Theorem 3.1. Let , and define, for each , , , and . It is easy to see that , , and for all . Note that, for each , and are monotone transformations of the initial variable . This implies that LNQD assumption is preserved by this construction in view of Lemma 2.6. Precisely, and are also LNQD sequences of zero mean random variables.For our purpose, it suffices to prove Using Lemma 2.8, the Hölder inequality, relation (3.2), and the second condition in (2.1) of the RCI() property of the sequence , we obtain This proves (3.4). To verify (3.3), using Lemma 2.7, we have Using the first condition of (2.1) of the RCI() property of the sequence , the last expression above clearly goes to 0 as , from and , thus completing the proof.

Remark 3.2. Let , and let be a LNQD sequence of random variables. If is RCI() for some , then .

Compared with Theorem 3.1, this result, whose proof can be completed by using Lemma 2.9, drops the maximum of the partial sum at the price of enlarging into .

Next we consider the case .

Theorem 3.3. Let , and let be a LNQD sequence of random variables. If satisfies then for any

Proof of Theorem 3.3. By Lemma 2.7 and the Hölder inequality, The proof is completed.

4. Strongly Residual Cesáro Alpha-Integrability and Complete Convergence of the Maximum of the Partial Sum

A sequence of random variables is said to converge completely to a constant if for any , In this case we write completely. This notion was given by Hsu and Robbins [16]. Note that the complete convergence implies the almost sure convergence in view of the Borel-Cantelli lemma.

The condition of SRCI() is a strong version of the condition of RCI(). In this section, we will show that each of the theorems in the previous section has a corresponding “strong” analogue in the sense of complete convergence.

Theorem 4.1. Let , and let be a LNQD sequence of random variables. If is SRCI for some , then

Proof of Theorem 4.1. For any , let be the integer such that . Observe that Hence it suffices to show that Let , , , and be defined as in the proof of Theorem 3.1. We first prove that completely; that is, Using Lemma 2.8, the Hölder inequality, relation (3.2), and the second condition in (2.1) of the RCI() property of the sequence , we have which implies (4.4).Next we show that completely; that is, By Lemma 2.7 and the Hölder inequality, In view of the first condition in (2.1) of the RCI() property of the sequence , we have The last series above converges since implies , and therefore (4.7) holds. This completes the proof.

For the case , we have the following result.

Theorem 4.2. Let , and let be a LNQD sequence of random variables. If satisfies then for any

Proof of Theorem 4.2. Let be defined as in the proof of Theorem 4.1. Proceeding in the proof of (4.3), we see that it suffices to show that Indeed by Lemma 2.7 and the Hölder inequality, In view of Lemma 2.10, from . Therefore (4.12) holds. The proof is completed.

Acknowledgments

Supported by the National Science Foundation of China (11061012), the Guangxi China Science Foundation (2010GXNSFA013120), and Innovation Project of Guangxi Graduate Education (2010105960202M32). We 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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