On Complete Moment Convergence of Weighted Sums for Arrays of Rowwise Negatively Associated Random Variables

The complete moment convergence of weighted sums for arrays of rowwise negatively associated random variables is investigated. Some sufficient conditions for complete moment convergence of weighted sums for arrays of rowwise negatively associated random variables are established. Moreover, the results of Baek et al. 2008 , are complemented. As an application, the complete moment convergence of moving average processes based on a negatively associated random sequences is obtained, which improves the result of Li et al. 2004 .


Introduction
Let {X n , n ≥ 1} be a sequence of random variables and, as usual, set S n n i 1 X i , n ≥ 1.When {X n , n ≥ 1} are independent and identically distributed i.i.d., Baum and Katz 1 proved the following remarkable result concerning the convergence rate of the tail probabilities P |S n | > n 1/p for any > 0.
There is an interesting and substantial literature of investigation apropos of extending the Baum-Katz theorems along a variety of different paths.One of these extensions is due Recently, Baum-Katz theorem is extended to the case of dependence random variables.Liang 3 obtained some general results on the complete convergence of weighted sums of negatively associated random variables.Li and Zhang 4 showed complete moment convergence for moving average processes under negative association as follows.
Theorem C see 4 .Suppose that X n ∞ i −∞ a i n Y i , n ≥ 1, where {a i , −∞ < i < ∞} is a sequence of real numbers with ∞ −∞ |a i | < ∞ and {Y i , −∞ < i < ∞} is a sequence of identically distributed and negatively associated random variables with EY 1 0, EY 2  1 < ∞.Let l x > 0 be a slowly varying function and Kuczmaszewska 5 proposed a very general result for complete convergence of rowwise negatively associated arrays of random variables which is stated in Theorem D.
Theorem D see 5 .Let {X ni , i ≥ 1, n ≥ 1} be an array of rowwise negatively associated random variables and let {a ni , i ≥ 1, n ≥ 1} be an array of real numbers.Let {b n , n ≥ 1} be an increasing sequence of positive integers and let {c n , n ≥ 1} be a sequence of positive real numbers.If for some q > 2, 0 < t < 2 and any > 0 the following conditions are fulfilled: for some r > 0, 1.5 a If α β 1 > 0 and there exists some δ > 0 such that α/r 1 < δ ≤ 2, and s max 1 In this paper, the authors take the inspiration in 5, 6 and discuss the complete moment convergence of weighted sums for arrays of rowwise negatively associated random variables by applying truncation methods, which extend the results of 5, 6 .As an application, the complete moment convergence of moving average processes based on a negatively associated random sequences is obtained, which extend the result of Li and Zhang 4 .
For the proof of the main results, we need to restate a few definitions and lemmas for easy reference.Throughout this paper, C will represent positive constants whom their value may change from one place to another.The symbol I A denotes the indicator function of A, x indicate the maximum integer not larger than x.For a finite set B, the symbol #B denotes the number of elements in the set B. Definition 1.1.A finite family of random variables {X i , 1 ≤ i ≤ n} is said to be negatively associated abbreviated to NA in the following , if for every pair disjoint subsets A and B of {1, 2, . . ., n} and any real nondecreasing coordinate-wise functions whenever the covariance exists.An infinite family of random variables The definition of negatively associated was introduced by Alam and Saxena 7 and was studied by Joag-Dev and Proschan 8 and Block et al. 9 .As pointed out and proved by Joag-Dev and Proschan, a number of well-known multivariate distributions possess the NA property.Negative association has found important and wide applications in multivariate statistical analysis and reliability.Many investigators discuss applications of negative association to probability, stochastic processes, and statistics.Definition 1.2.A sequence {X n , n ≥ 1} of random variables is said to be stochastically dominated by a random variable X write {X i } ≺ X if there exists a constant C, such that The following lemma is a well-known result.
Lemma 1.3.Let the sequence {X n , n ≥ 1} of random variables be stochastically dominated by a random variable X.Then for any p > 0, x > 0 1.9 Definition 1.4.A real-valued function l x , positive and measurable on A, ∞ for some A > 0, is said to be slowly varying if lim x → ∞ l xλ /l x 1 for each λ > 0.
By the properties of slowly varying function, we can easily prove the following two lemmas.Here we omit the details of the proof.Lemma 1.5.Let l x > 0 be a slowly varying function as x → ∞.
n k n r l n ≤ Ck r 1 l k for any r < −1 and positive integer k.
Lemma 1.6.Let X be a random variable and let l x > 0 be a slowly varying function as x → ∞.
The following lemma will play an important role in the proof of our main results.The proof is according to Shao 10 .Lemma 1.7.Let {X i , 1 ≤ i ≤ n} be a sequence of NA random variables with mean zero and 1.10 By monotone convergence and 1.10 , we have the following lemma.
Lemma 1.8.Let {X i , i ≥ 1} be a sequence of NA random variables with mean zero and Using Lemma 1.4, Lemma 1.5, and Theorem 2.11 in Sung 11 , we obtain the following lemmas.
Lemma 1.9.Let {X ni , 1 ≤ i ≤ n, n ≥ 1} be an array of rowwise NA random variables with E|X ni | < ∞ for 1 ≤ i ≤ n, n ≥ 1.Let {b n , n ≥ 1} be a sequence of real numbers.If for some q ≥ 2, the following conditions are fulfilled: Let {b n , n ≥ 1} be a sequence of real numbers.If for some q > 2, the following conditions are fulfilled:

Main Results
Now we state our main results.The proofs will be given in Section 3.
ii If α β 1 0, assume also l x ≤ Cl y for all 0 < x < y.

2.10
Thus, we improve the results of Baek et al. 6 to supreme value of partial sums.

2.11
Corollary 2.5.Under the conditions of Theorem 2.2, Corollary 2.6.Let {X ni , 1 ≤ i ≤ n, n ≥ 1} be an array of rowwise NA random variables with EX ni 0 and stochastically dominated by a random variable X. Suppose that l x > 0 is a slowing varying function.
and {Y i , −∞ < i < ∞} is a NA random sequence with EY i 0 and is stochastically dominated by a random variable Y .Let l x be a slowly varying function.

2.16
Remark 2.8.Theorem 2.7 obtains the result about the complete moment convergence of moving average processes based on an NA random sequence with different distributions.The result of Li and Zhang 4 is a special case of Theorem 2.7 1 .Moreover, our result covers the case of r t, which was not considered by Li and Zhang.

Proofs of the Main Results
Proof of Theorem 2.1.Since a ni a ni − a − ni , where a ni max a ni , 0 and a − ni max −a ni , 0 , we have So, without loss of generality, we can assume a ni > 0. From 2.1 and 2.2 , without loss of generality, we assume sup 1≤i≤n Put b n n β l n , n 1, 2, . . . in Lemma 1.9.Noting that α β 1 > 0, by Lemma 1.3 and Lemma 1.7, we have

3.3
Since α < r, we can take some t such that 1 α/r < t ≤ min s, 2 .Observe that

3.5
By 3.3 and Lemma 1.3, we have

3.6
Set Hence, we have 3.9 Choosing q large enough such that α β r − rq < −1, we obtain by Lemma 1.6 and 3.8 that

3.11
From 3.6 , 3.9 , 3.10 , and 3.11 , we know that 3.12 By 3.3 , 3.5 , and 3.12 , we see that a , b , and c in Lemma 1.9 with X ni replaced by a ni X ni are fulfilled.Since {a ni X ni , 1 ≤ i ≤ n, n ≥ 1} is also an array of rowwise NA random variables, by Lemma 1.9, we complete the proof of 2.3 .
Next, we prove 2.4 .If α β 1 0, then k n 1 n α β ≤ C log 1 n .Similarly for the proof of 3.3 , noting that l x ≤ Cl y , 0 < x < y, we have

3.14
Thus, for q 2, a , b , and c in Lemma

1 1 ≤ C ∞ n 1 n
E|a ni X| 2 I |a ni X| ≤ α β r−2r l n E|X| 2 I |X| ≤ 2n r C ∞ n 1 Theorem E see 6 .Let {X ni , i ≥ 1, n ≥ 1} be an array of rowwise negatively associated random variables with EX ni 0 and P {|X ni | > x} ≤ CP {|X| > x} for all n, i and x ≥ 0. Suppose that β ≥ −1, and that {a ni , i ≥ 1, n ≥ 1} is an array of constants such that Baek et al. 6discussed complete convergence of weighted sums for arrays of rowwise negatively associated random variables and obtained the following results.
be an array of rowwise NA random variables with EX ni 0 and stochastically dominated by a random variable X. Suppose that l x > 0 is a slowing varying function and that {a ni , 1 ≤ i ≤ n, n ≥ 1} is an array of constants such that n i 1 |a ni | O n α for some α ∈ 0, r .2.2Journal of Probability and Statisticsi If α β 1 > 0 and there exists some δ > 0 such that α/r 1 < δ ≤ 2, and s max 1 α β 1 /r , δ , then 1.9 with X ni replaced by a ni X ni are fulfilled.So 2.4 holds.Proof of Theorem 2.2.By Lemma 1.10, the rest of the proof is similar to that of Theorem 2.1 and is omitted.Proof of Corollary 2.6.By applying Theorem 2.1, taking β p − 2, a ni n −1/t for 1 ≤ i ≤ n, and a ni 0 for i > n, then we obtain 2.13 .Similarly, taking β −1, a ni n −1/t for 1 ≤ i ≤ n, and a ni 0 for i > n, we obtain 2.14 by Theorem 2.1.Proof of Theorem 2.7.Let X ni Y i anda ni n −1/t n j 1 a i j for all n ≥ 1, −∞ < i < ∞.Since ∞ −∞ |a i | < ∞, we have sup i |a ni | O n −1/t and ∞ i −∞ |a ni | O n 1−1/t .By applying Corollary 2.5, taking β r/t − 2, r 1/t, α 1 − 1/t , we obtain