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

Abstract

Let { [subscript] X n i [/subscript] ; i ...5; 1 , n ...5; 1 } 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).

(ProQuest: ... denotes non-US-ASCII text omitted.)

Academic Editor:Baodong Zheng

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China

Received 24 April 2014; Accepted 3 August 2014; 20 August 2014

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.

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 { _{X i} ; 1 ...4; i ...4; n } is said to be NA if for every pair of disjoint subsets _{A 1} and _{A 2} of { 1,2 , ... , n } , [figure omitted; refer to PDF] whenever _{f 1} and _{f 2} are nondecreasing functions such that the covariance exists. An infinite collection of random variables { _{X i} ; i ...5; 1 } is NA if every finite subcollection is NA.

An array of random variables { _{X n i} ; i ...5; 1 , n ...5; 1 } is called rowwise NA random variables if for every n ...5; 1 , { _{X n i} ; i ...5; 1 } is a sequence of NA random variables.

Definition 2.

A finite collection of random variables { _{X i} ; 1 ...4; i ...4; n } is said to be NOD if [figure omitted; refer to PDF] for all _{x 1} , _{x 2} , ... , _{x n} ∈ R . An infinite collection of random variables { _{X i} ; i ...5; 1 } is said to be NOD if every finite subcollection is NOD.

An array of random variables { _{X n i} ; i ...5; 1 , n ...5; 1 } is called rowwise NOD random variables if for every n ...5; 1 , { _{X n i} ; i ...5; 1 } 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 { _{X n} ; n ...5; 1 } be a sequence of independent and identically distributed (i.i.d.) random variables and let { _{a n i} ; i ...5; 1 , n ...5; 1 } 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 ^{∑ i = 1 n} _{a n i X i} 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 { X , _{X n} ; n ...5; 1 } be a sequence of identically distributed NA random variables, and let { _{a n i} ; 1 ...4; i ...4; n , n ...5; 1 } be an array of real constants satisfying [figure omitted; refer to PDF] for some 0 < α ...4; 2 . Suppose that E X = 0 when 1 < α ...4; 2 . If [figure omitted; refer to PDF] then, for _{b n} = ^{n 1 / α} ( log ... ... n ^{) 1 / γ} , [figure omitted; refer to PDF]

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

Theorem B.

Let { _{X n i} ; i ...5; 1 , n ...5; 1 } be an array of rowwise NOD random variables which is stochastically dominated by a random variable X and let { _{a n i} ; 1 ...4; i ...4; n , n ...5; 1 } be an array of real constants. Assume that there exist some δ with 0 < δ < 1 and some α with 0 < α < 2 such that ^{∑ i = 1 n} | _{a n i} ^{| α} = O ( ^{n δ} ) and assume further that E _{X n i} = 0 if 1 < α < 2 . If for some h > 0 and γ > 0 such that (4), then [figure omitted; refer to PDF] where p ...5; 1 / α and _{b n} = ^{n 1 / α} ( log ... ... n ^{) 1 / γ} .

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 { _{X n} ; n ...5; 1 } be a sequence of NOD random variables which is stochastically dominated by a random variable X and let { _{a n i} ; i ...5; 1 , n ...5; 1 } be a triangular array of real constants such that _{a n i} = 0 for i > n . Let [figure omitted; refer to PDF] where β = max ... ( α , γ ) for some 0 < α ...4; 2 , γ > 0 , and α ...0; γ . Assume that E _{X n} = 0 for 1 < α ...4; 2 and E ^{| X | β} < ∞ . Then, [figure omitted; refer to PDF] where _{b n} = ^{n 1 / α} ( log ... ... n ^{) 1 / γ} .

As Huang and Wang [12] pointed out, Theorem C partially extends only the case of α > γ of Theorems A and B. They left an open problem whether the case of α = γ 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 { _{X n i} ; i ...5; 1 , n ...5; 1 } is said to be stochastically dominated by a random variable X if there exists a positive constant C such that [figure omitted; refer to PDF] for all t ...5; 0 , i ...5; 1 , and n ...5; 1 .

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 { _{X n i} ; i ...5; 1 , n ...5; 1 } be an array of rowwise NOD random variables which is stochastically dominated by a random variable X and let { _{a n i} ; 1 ...4; i ...4; n , n ...5; 1 } be an array of real constants satisfying ^{∑ i = 1 n} | _{a n i} ^{| α} = O ( n ) for some 0 < α ...4; 2 . Assume further that E _{X n i} = 0 for 1 < α ...4; 2 and E | X ^{| α} log ... ( 1 + | X | ) < ∞ . Then, [figure omitted; refer to PDF] where _{b n} = ^{n 1 / α} ( log ... ... n ^{) 1 / α} .

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

Corollary 5.

Let { _{X n} ; n ...5; 1 } be a sequence of NOD random variables which is stochastically dominated by a random variable X and let { _{a n i} ; 1 ...4; i ...4; n , n ...5; 1 } be an array of real constants satisfying ^{∑ i = 1 n} | _{a n i} ^{| α} = O ( n ) for some 0 < α ...4; 2 . Assume further that E _{X n} = 0 for 1 < α ...4; 2 and E | X ^{| α} log ... ( 1 + | X | ) < ∞ . Then, [figure omitted; refer to PDF] where _{b n} = ^{n 1 / α} ( log ... ... n ^{) 1 / α} .

Remark 6.

In Theorem 4 and Corollary 5, we consider the case of α = γ for 0 < α ...4; 2 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 [figure omitted; refer to PDF] 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 { _{X i} ; 1 ...4; i ...4; n } be a sequence of NOD random variables, and let { _{f i} ; 1 ...4; i ...4; n } be a sequence of Borel functions all of which are monotone nondecreasing (or all are monotone nonincreasing). Then, { _{f i} ( _{X i} ) ; 1 ...4; i ...4; n } is a sequence of NOD random variables.

Lemma 8 (see Asadian et al. [3]).

Let M ...5; 2 and let { _{X n} ; n ...5; 1 } be a sequence of NOD random variables with E _{X n} = 0 and E | _{X n} ^{| M} < ∞ for all n ...5; 1 . Then, there exists a positive constant C = C ( M ) depending only on M such that, for all n ...5; 1 , [figure omitted; refer to PDF]

Lemma 9.

Let { _{X n} ; n ...5; 1 } be a sequence of random variables which is stochastically dominated by a random variable X . For any u > 0 and t > 0 , the following two statements hold: [figure omitted; refer to PDF] where _{C 1} and _{C 2} are positive constants.

Lemma 10 (see Sung [15]).

Let X be a random variable and let { _{a n i} ; 1 ...4; i ...4; n , n ...5; 1 } be an array of real constants satisfying ^{∑ i = 1 n} | _{a n i} ^{| α} = O ( n ) for some α > 0 . Let _{b n} = ^{n 1 / α} ( log ... ... n ^{) 1 / γ} for some γ > 0 . Then, [figure omitted; refer to PDF]

Lemma 11 (see Sung [19]).

Let X be a random variable and let { _{a n i} ; 1 ...4; i ...4; n , n ...5; 1 } be an array of real constants satisfying _{a n i} = 0 or | _{a n i} | > 1 and ^{∑ i = 1 n} | _{a n i} ^{| α} = O ( n ) for some α > 0 . Let _{b n} = ^{n 1 / α} ( log ... ... n ^{) 1 / α} . If q > α , then [figure omitted; refer to PDF]

Throughout this paper, let I ( A ) be the indicator function of the set A . C denotes a positive constant, which may be different in various places and _{a n} = O ( _{b n} ) stands for _{a n} ...4; C _{b n} .

Proof of Theorem 4.

Without loss of generality, suppose that ^{∑ i = 1 n | _{a n i} | α} ...4; C n and _{a n i} ...5; 0 , for all 1 ...4; i ...4; n , n ...5; 1 . For fixed n ...5; 1 , define [figure omitted; refer to PDF] Denote [figure omitted; refer to PDF] It is easily seen that, for all [straight epsilon] > 0 , [figure omitted; refer to PDF] which implies that [figure omitted; refer to PDF] First, we will prove that [figure omitted; refer to PDF] Actually, for 0 < α ...4; 1 , by (14) of Lemma 9, Markov inequality, and E | X ^{| α} log ... ( 1 + | X | ) < ∞ , we have that [figure omitted; refer to PDF] Next, for 1 < α ...4; 2 , by E _{X n i} = 0 , (15) of Lemmas 9 and 10, Markov inequality, and E | X ^{| α} log ... ( 1 + | X | ) < ∞ , we also have that [figure omitted; refer to PDF] From the above statements, we can get (22) immediately. Hence, for n large enough, [figure omitted; refer to PDF] To prove (10), it is sufficient to show that [figure omitted; refer to PDF] It follows from Lemma 10 and E | X ^{| α} log ... ( 1 + | X | ) < ∞ that [figure omitted; refer to PDF] For fixed n ...5; 1 , it is easily seen that { ^{X i ( n )} - E ^{X i ( n )} , i ...5; 1 , n ...5; 1 } 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 M > 2 ) that [figure omitted; refer to PDF]

It follows from Lemma 10, (14) of Lemma 9, and Markov inequality that [figure omitted; refer to PDF] From Lemma 10 and E | X ^{| α} log ... ( 1 + | X | ) < ∞ , we can obtain that [figure omitted; refer to PDF]

For fixed n > 1 , we divide { a n i , 1 ...4; i ...4; n } into three subsets { a n i : | a n i | ...4; 1 / ( log ... ... n ) m } , { a n i : 1 / ( log ... ... n ) m < | a n i | ...4; 1 } , and { a n i : | a n i | > 1 } , where m = ( 1 / ( M - α ) ) . Then, [figure omitted; refer to PDF] By Lemma 11 and E | X | α log ... ( 1 + | X | ) < ∞ again, it follows that [figure omitted; refer to PDF] Noting that ∑ i : | a n i | ...4; 1 / ( log ... ... n ) m | a n i | α ...4; C n ( log ... ... n ) - m α , for M > α and fixed n > 1 , we have that [figure omitted; refer to PDF] Noting that ∑ i : 1 / ( log ... ... n ) m < | a n i | ...4; 1 | a n i | M ...4; C n and m = 1 / ( M - α ) , for M > 2 , 0 < α ...4; 2 , we have that [figure omitted; refer to PDF] Finally, we will prove that [figure omitted; refer to PDF] Hence, by C r inequality, Markov inequality, Lemmas 9-11, and E | X | α log ... ( 1 + | X | ) < ∞ , we have that [figure omitted; refer to PDF] Therefore, the desired result (10) follows from the above statements. This completes the proof of Theorem 4.

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).

Conflict of Interests

The authors declare that they have no conflict of interests.

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