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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 425969, 13 pages
http://dx.doi.org/10.1155/2012/425969
Research Article

Equivalent Conditions of Complete Convergence for Weighted Sums of Sequences of Negatively Dependent Random Variables

School of Mathematics and Computer Science, Anhui Normal University, Wuhu 241003, China

Received 29 August 2012; Accepted 14 October 2012

Academic Editor: Josef Diblík

Copyright © 2012 Mingle Guo. 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

The complete convergence for weighted sums of sequences of negatively dependent random variables is investigated. By applying moment inequality and truncation methods, the equivalent conditions of complete convergence for weighted sums of sequences of negatively dependent random variables are established. These results not only extend the corresponding results obtained by Li et al. (1995), Gut (1993), and Liang (2000) to sequences of negatively dependent random variables, but also improve them.

1. Introduction

In many stochastic model, the assumption that random variables are independent is not plausible. Increases in some random variables are often related to decreases in other random variables, so an assumption of negatively dependence is more appropriate than an assumption of independence.

Lehmann [1] introduced the notion of negatively quadrant dependent (NQD) random variables in the bivariate case.

Definition 1.1. Random variables and are said to be NQD if for all . A sequence of random variables is said to be pairwise NQD if for all , and are NQD.

It is important to note that (1.1) is equivalent to for all . However, (1.1) and (1.2) are not equivalent for a collection of three or more random variables. Consequently, the definition of NQD was extended to the multivariate case by Ebrahimi and Ghosh [2].

Definition 1.2. A finite family of random variables is said to be negatively dependent (ND) if for all real numbers , An infinite family of random variables is ND if every finite subfamily is ND.

Negative dependence has been very useful in reliability theory and applications. Since the paper of Ebrahimi and Ghosh [2] appeared, Taylor et al. [3, 4] studied the laws of large numbers for arrays of rowwise ND random variables, Ko and Kim [5] and Ko et al. [6] investigated the strong laws of large numbers for weighted sums of ND random variables. Volodin [7] obtained the Kolmogorov exponential inequality for ND random variables, Amini and Bozorgnia [8] studied the complete convergence for ND random variable sequences, Volodin et al. [9] obtained the convergence rates in the form of a Baum-Katz, and Wu [10] investigated complete convergence for weighted sums of sequences of negatively dependent random variables.

The concept of negatively associated random variables was introduced by Alam and Saxena [11] and carefully studied by Joag-Dev and Proschan [12].

Definition 1.3. A finite family of random variables is said to be negatively associated (NA), if for every pair disjoint subset and of and any real nondecreasing coordinate-wise functions on and on , whenever the covariance exists. An infinite family of random variables is NA if every finite subfamily is NA.

As pointed out and proved by Joag-Dev and Proschan [12], a number of well-known multivariate distributions posses the NA property, such as multinomial, convolution of unlike multinomial, multivariate hypergeometric, Dirichlet, permutation distribution, random sampling without replacement, and joint distribution of ranks.

Obviously, NA implies ND from the definition of NA and ND. But ND does not imply NA, so ND is much weaker than NA. Hence, the extending the limit properties of independent random variables to the case of ND random variables is highly desirable and considerably significant in the theory and application.

The concept of complete convergence of a sequence of random variables was introduced by Hsu and Robbins [13] as follows. A sequence of random variables is said to converge completely to a constant if

In view of the Borel-Cantelli lemma, the complete convergence implies almost sure convergence. Therefore, the complete convergence is very important tool in establishing almost sure convergence. When is independent and identically distributed (i.i.d), Baum and Katz [14] proved the following remarkable result concerning the convergence rate of the tail probabilities for any .

Theorem A. Let and . Then, if and only if , where whenever .

There is an interesting and substantial literature of investigation of extending the Baum-Katz Theorem along a variety of different paths. Since partial sums are a particular case of weighted sums and the weighted sums are often encountered in some actual questions, the complete convergence for the weighted sums seems more important. Li et al. [15] discussed the complete convergence for independent weighted sums. Gut [16] discussed complete convergence of Cesàro means of i.i.d random variables. Liang [17] extended the conclusions of Li et al. [15] and Gut [16] to NA random variables and obtained the following results.

Theorem B. Let be a sequence of identically distributed NA random variables and let . Assume and is a triangular array of real numbers such that for all . Then, the following are equivalent:

Theorem C. Let be a sequence of identically distributed NA random variables and let . Then, the following are equivalent:

where , and .

In the current work, we study the complete convergence for ND random variables. Equivalent conditions of complete convergence for weighted sums of sequences of ND random variables are established. As a result, we not only promote and improve the results of Liang [17] for NA random variables to ND random variables without necessarily imposing any extra conditions, but also relax the range of .

For the proofs of the main results, we need to restate a few lemmas for easy reference. Throughout this paper, The symbol denotes a positive constant which is not necessarily the same one in each appearance and denotes the indicator function of . Let denote that there exists a constant such that for sufficiently large , and let mean and . Also, let denote .

Lemma 1.4 (see [11]). Let be a sequence of ND random variables and let be a sequence of Borel functions all of which are monotone increasing (or all are monotone decreasing). Then, is still a sequence of ND random variables.

Lemma 1.5 (see [18]). Let be a sequence of ND random variables, . Then, where depends only on .

Lemma 1.6 (see [10]). Let be a sequence of ND random variables. Then, there exists a positive constant such that for any and all ,

By Lemma 1.2 and Theorem 3 in [19], we can obtain the following lemma.

Lemma 1.7. Let be a sequence of ND random variables, . Then, where depends only on .

By using Fubini’s theorem, the following lemma can be easily proved. Here, we omit the details of the proof.

Lemma 1.8. Let be a random variable, then (i) for any and ;(ii) for any and .

2. Main Results

Now we state our main results. The proofs will be given in Section 3.

Theorem 2.1. Let be a sequence of identically distributed ND random variables, and suppose that for . Assume that is a triangular array of real numbers. Then, the following are equivalent:

Theorem 2.2. Let be a sequence of identically distributed ND random variables, and suppose that for . Assume that is a triangular array of real numbers. Then, the following are equivalent:

Remark 2.3. Since NA random variables are a special case of ND random variables, taking in Theorem 2.1, we obtain the result of Liang [17]. Thus, we not only promote and improve the results of Liang [17] for NA random variables to ND random variables without necessarily imposing any extra conditions, but also relax the range of .

Remark 2.4. Taking in Theorem 2.1, we improve the result of Baum and Katz [14].

Corollary 2.5. Let be a sequence of identically distributed ND random variables, , and . Let . Then, the following are equivalent:

3. Proofs of the Main Results

Proof of Theorem 2.1. First, we prove (2.1)  (2.2). Note that , where and . Thus, to prove (2.2), it suffices to show that So, without loss of generality, we can assume that . Choose being small enough and sufficient large integer . Let, for every , Obviously, . Note that Thus, in order to prove (2.2), it suffices to show that By the definition of , we see that . Since , by Lemma 1.8, we have Therefore, by (2.1), . From the definition of , we know . By using the definition of ND family, we have Since (2.1) implies , by Markov's inequality and (3.6), we obtain Noting that , we can choose being small enough and sufficient large integer such that and . Thus, by (3.7), we get . Similarly, we can obtain . In order to estimate , we first verify that Note that (2.1) implies and . When , noting that and , by Hölder’s inequality we have When , noting that , by choosing being small enough such that , we have Therefore, to prove , it suffices to prove that Note that is still ND by Lemma 1.4. Using Markov's inequality, inequality, and Lemma 1.7, we get for a suitably large , which will be determined later, Choosing sufficient large such that , we have When , (2.1) implies . Noting that , we can choose sufficient large such that . Then, When , choosing sufficient large such that , we have Thus, by (3.13), (3.14), and (3.15), we have .
Now, we proceed to prove (2.2)  (2.1). Since , by (2.2), we have Next, we shall prove In fact, when , (3.16) obviously implies (3.17). When , noting that , by (3.16), we obtain Obviously, is a nondecreasing sequence of nonnegative random variables. By (3.18), we have Noting that , by (3.19), we have Now, for each , let be such that . Then, for all Noting that , we have Therefore, by (3.21) and (3.22), we get that (3.17) holds.
Thus, by (3.17) and Lemma 1.6, we have Now (3.16) and (3.23) yield By the process of proof of (3.5), we see that (3.24) is equivalent to (2.1).

Proof of Theorem 2.2. The proof is similar to that of Theorem 2.1 and is omitted.

Proof of Corollary 2.5. Put . Note that, for , . Therefore, for , we obtain ,. Thus, letting in Theorem 2.2, we can conclude that (2.5) is equivalent to (2.6).

Acknowledgments

The research is supported by the National Natural Science Foundation of China (nos. 11271020 and 11201004), the Key Project of Chinese Ministry of Education (nos. 211077) and the Anhui Provincial Natural Science Foundation (nos. 10040606Q30 and 1208085MA11).

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