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

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).


Introduction
Firstly, let us recall the definitions of negatively associated (NA) random variables and NOD random variables as follows.
An array of random variables {  ;  ≥ 1,  ≥ 1} is called rowwise NA random variables if for every  ≥ 1, {  ;  ≥ 1} is a sequence of NA random variables.Definition 2. A finite collection of random variables {  ; 1 ≤  ≤ } is said to be NOD if for all  1 ,  2 , . . .,   ∈ R.An infinite collection of random variables {  ;  ≥ 1} is said to be NOD if every finite subcollection is NOD.
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.
Cai [17] proved the following complete convergence result for weighted sums of NA random variables.
Theorem A. Let {,   ;  ≥ 1} be a sequence of identically distributed NA random variables, and let {  ; 1 ≤  ≤ ,  ≥ 1} be an array of real constants satisfying then, for   =  1/ (log ) Wang et al. [11] extended the above result of Cai [17] to arrays of rowwise NOD random variables as follows.
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 {  ;  ≥ 1} be a sequence of NOD random variables which is stochastically dominated by a random variable  and let {  ;  ≥ 1,  ≥ 1} be a triangular array of real constants such that   = 0 for  > .Let where  = max(, ) for some 0 <  ≤ 2,  > 0, and  ̸ = .Assume that   = 0 for 1 <  ≤ 2 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 {  ;  ≥ 1,  ≥ 1} is said to be stochastically dominated by a random variable  if there exists a positive constant  such that for all  ≥ 0,  ≥ 1, and  ≥ 1.

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 {  ;  ≥ 1,  ≥ 1} be an array of rowwise NOD random variables which is stochastically dominated by a random variable  and let {  ; 1 ≤  ≤ ,  ≥ 1} be an array of real constants satisfying where Similar to the proof of Theorem 4, we can obtain the following result for NOD random variable sequences.

Proofs
In order to prove our main results, the following lemmas are needed.
where  1 and  2 are positive constants.
Hence, for  large enough, To prove (10), it is sufficient to show that It follows from Lemma 10 and For fixed  ≥ 1, it is easily seen that { ()  − ()  ,  ≥ 1,  ≥ 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 It follows from Lemma 10,(14) of Lemma 9, and Markov inequality that Therefore, the desired result (10) follows from the above statements.This completes the proof of Theorem 4.