On Complete Convergence for Weighted Sums of Arrays of Dependent Random Variables

and Applied Analysis 3 in probability. If E|X| log|X| < ∞, for 1 α β 0, E|X| 1 α β /γ < ∞, for 1 α β > 0, 1.8 then 1.5 holds. In this paper, we extend Theorem 1.3 to negatively associated and negatively dependent random variables. We also obtain similar results for sequences of φ-mixing and ρ∗-mixing random variables. Our results improve and generalize the results of Baek et al. 4 , Kuczmaszewska 6 , and Wang et al. 7 . Throughout this paper, the symbol C denotes a positive constant which is not necessarily the same one in each appearance. It proves convenient to define logx max{1, lnx}, where lnx denotes the natural logarithm. 2. Preliminaries In this section, we present some background materials which will be useful in the proofs of our main results. The following lemma is well known, and its proof is standard. Lemma 2.1. Let {Xn, n ≥ 1} be a sequence of random variables which are stochastically dominated by a random variable X. For any α > 0 and b > 0, the following statements hold: i E|Xn|I |Xn| ≤ b ≤ C{E|X|I |X| ≤ b bP |X| > b }, ii E|Xn|I |Xn| > b ≤ CE|X|I |X| > b . Lemma 2.2 Sung 8 . Let X be a random variable with E|X| < ∞ for some r > 0. For any t > 0, the following statements hold: i ∑∞ n 1n −1−tδE|X|r I |X| ≤ n ≤ CE|X| for any δ > 0, ii ∑∞ n 1n −1 tδE|X|r−δI |X| > n ≤ CE|X| for any δ > 0 such that r − δ > 0, iii ∑∞ n 1n −1 P |X| > n ≤ CE|X| . The Rosenthal-type inequality plays an important role in establishing complete convergence. The Rosenthal-type inequalities for sequences of dependent random variables have been established by many authors. The concept of negatively associated random variables was introduced by Alam and Saxena 9 and carefully studied by Joag-Dev and Proschan 10 . A finite family of random variables {Xi, 1 ≤ i ≤ n} is said to be negatively associated if for every pair of disjoint subsets A and B of {1, 2, . . . , n}, Cov ( f1 Xi, i ∈ A , f2 ( Xj, j ∈ B )) ≤ 0, 2.1 whenever f1 and f2 are coordinatewise increasing and the covariance exists. An infinite family of random variables is negatively associated if every finite subfamily is negatively associated. 4 Abstract and Applied Analysis The following lemma is a Rosenthal-type inequality for negatively associated random variables. Lemma 2.3 Shao 11 . Let {Xn, n ≥ 1} be a sequence of negatively associated random variables with EXn 0 and E|Xn| < ∞ for some q ≥ 2 and all n ≥ 1. Then there exists a constant C > 0 depending only on q such that E ⎛ ⎝max 1≤j≤n ∣∣∣∣ j ∑ i 1 Xi ∣∣∣∣ q ⎞ ⎠ ≤ C ⎧ ⎨ ⎩ n ∑ i 1 E|Xi| ( n ∑ i 1 EX2 i )q/2⎬ ⎭ 2.2 The concept of negatively dependent random variables was given by Lehmann 12 . A finite family of random variables {X1, . . . , Xn} is said to be negatively dependent or negatively orthant dependent if for each n ≥ 2, the following two inequalities hold: P X1 ≤ x1, . . . , Xn ≤ xn ≤ n ∏ i 1 P Xi ≤ xi , P X1 > x1, . . . , Xn > xn ≤ n ∏