A Note on the Rate of Strong Convergence for Weighted Sums of Arrays of Rowwise Negatively Orthant Dependent Random Variables

Abstract

Let be an array of rowwise negatively orthant dependent (NOD) random variables. The authors discuss the rate of strong convergence for weighted sums of arrays of rowwise NOD random variables and solve an open problem posed by Huang and Wang (2012).

1. Introduction

Firstly, let us recall the definitions of negatively associated (NA) random variables and NOD random variables as follows.

Definition 1. A finite collection of random variables is said to be NA if for every pair of disjoint subsets and of , whenever and are nondecreasing functions such that the covariance exists. An infinite collection of random variables is NA if every finite subcollection is NA.

An array of random variables is called rowwise NA random variables if for every is a sequence of NA random variables.

Definition 2. A finite collection of random variables is said to be NOD if for all . An infinite collection of random variables is said to be NOD if every finite subcollection is NOD.

An array of random variables is called rowwise NOD random variables if for every is a sequence of NOD random variables.

The concepts of NA and NOD random variables were introduced by Joag-Dev and Proschan [1]. Obviously, independent random variables are NOD, and NA implies NOD from the definition of NA and NOD, but NOD does not imply NA. So, NOD is much weaker than NA. Because of the wide applications of NOD random variables, the notion of NOD random variables has been received more and more attention recently. Many applications have been found. We can refer to Volodin [2], Asadian et al. [3], Amini et al. [4, 5], Kuczmaszewska [6], Zarei and Jabbari [7], Wu and Zhu [8], Wu [9], Sung [10], Wang et al. [11], Huang and Wang [12], and so forth. Hence, it is very significant to study limit properties of this wider NOD random variables in probability theory and practical applications.

Let be a sequence of independent and identically distributed (i.i.d.) random variables and let be an array of real constants. As Bai and Cheng [13] remarked, many useful linear statistics, for example, least-squares estimators, nonparametric regression function estimators, and jackknife estimates, are based on weighted sums of i.i.d. random variables. In this respect, the strong convergence for weighted sums has been studied by many authors (see, e.g., Bai and Cheng [13]; Cuzick [14]; Sung [15]; Tang [16]; etc.).

Cai [17] proved the following complete convergence result for weighted sums of NA random variables.

Theorem A. Let be a sequence of identically distributed NA random variables, and let be an array of real constants satisfying for some . Suppose that when . If then, for ,

Wang et al. [11] extended the above result of Cai [17] to arrays of rowwise NOD random variables as follows.

Theorem B. Let be an array of rowwise NOD random variables which is stochastically dominated by a random variable and let be an array of real constants. Assume that there exist some with and some with such that and assume further that if . If for some and such that (4), then where and .

Recently, Huang and Wang [12] partially extended the corresponding theorems of Cai [17] and Wang et al. [11] to NOD random variables under a mild moment condition.

Theorem C. Let be a sequence of NOD random variables which is stochastically dominated by a random variable and let be a triangular array of real constants such that for . Let where for some , , and . Assume that for and . Then, where .

As Huang and Wang [12] pointed out, Theorem C partially extends only the case of of Theorems A and B. They left an open problem whether the case of of Theorem C holds for NOD random variables.

The main purpose of this paper is to further study strong convergence for weighted sums of NOD random variables and to obtain the rate of strong convergence for weighted sums of arrays of rowwise NOD random variables under a suitable moment condition. We solve the above problem posed by Huang and Wang [12].

We will use the following concept in this paper.

Definition 3. An array of random variables is said to be stochastically dominated by a random variable if there exists a positive constant such that for all , and .

2. Main Results

Now, we will present the main results of this paper; the detailed proofs will be given in the next section.

Theorem 4. Let be an array of rowwise NOD random variables which is stochastically dominated by a random variable and let be an array of real constants satisfying for some . Assume further that for and . Then, where .

Similar to the proof of Theorem 4, we can obtain the following result for NOD random variable sequences.

Corollary 5. Let be a sequence of NOD random variables which is stochastically dominated by a random variable and let be an array of real constants satisfying for some . Assume further that for and . Then, where .

Remark 6. In Theorem 4 and Corollary 5, we consider the case of for and obtain some strong convergence results for arrays of rowwise NOD random variables and NOD random variable sequences without assumption of identical distribution. The main result settles the open problem posed by Huang and Wang [12]. In addition, it is still an open problem whether holds true under the same moment condition of Theorem 4.

3. Proofs

In order to prove our main results, the following lemmas are needed.

Lemma 7 (see Bozorgnia et al. [18]). Let be a sequence of NOD random variables, and let be a sequence of Borel functions all of which are monotone nondecreasing (or all are monotone nonincreasing). Then, is a sequence of NOD random variables.

Lemma 8 (see Asadian et al. [3]). Let and let be a sequence of NOD random variables with and for all . Then, there exists a positive constant depending only on such that, for all ,

Lemma 9. Let be a sequence of random variables which is stochastically dominated by a random variable . For any and , the following two statements hold: where and are positive constants.

Lemma 10 (see Sung [15]). Let be a random variable and let be an array of real constants satisfying for some . Let for some . Then,

Lemma 11 (see Sung [19]). Let be a random variable and let be an array of real constants satisfying or and for some . Let . If , then

Throughout this paper, let be the indicator function of the set . denotes a positive constant, which may be different in various places and stands for .

Proof of Theorem 4. Without loss of generality, suppose that and , for all . For fixed , define Denote It is easily seen that, for all , which implies that First, we will prove that Actually, for , by (14) of Lemma 9, Markov inequality, and , we have that Next, for , by , (15) of Lemmas 9 and 10, Markov inequality, and , we also have that From the above statements, we can get (22) immediately. Hence, for large enough, To prove (10), it is sufficient to show that It follows from Lemma 10 and that For fixed , it is easily seen that is still a sequence of NOD random variables with mean zero by Lemma 7. Hence, it follows from (14) of Lemmas 9 and 8 and Markov inequality (for ) that
It follows from Lemma 10, (14) of Lemma 9, and Markov inequality that From Lemma 10 and , we can obtain that
For fixed , we divide into three subsets , , and , where . Then, By Lemma 11 and again, it follows that Noting that , for and fixed , we have that Noting that and , for , we have that Finally, we will prove that Hence, by inequality, Markov inequality, Lemmas 9–11, and , we have that Therefore, the desired result (10) follows from the above statements. This completes the proof of Theorem 4.

Conflict of Interests

The authors declare that they have no conflict of interests.

Acknowledgments

The authors are most grateful to the referees and to the editor Professor Baodong Zheng for their valuable suggestions and some helpful comments which greatly improved the clarity and readability of this paper. This paper is partially supported by the National Nature Science Foundation of China (71271042), the Fundamental Research Funds for the Central Universities of China (ZYGX2012J119), the Nature Science Foundation of Guangxi Province (2014GXNSFBA118006, 2013GXNSFDA019001), and the Guangxi Provincial Scientific Research Projects (201204LX157, 2013YB104).