Abstract
A complete convergence result for an array of rowwise independent mean zero random variables was established by Kruglov et al. (2006). This result was partially extended to negatively associated and negatively dependent mean zero random variables by Chen et al. (2007) and Dehua et al. (2011), respectively. In this paper, we obtain complete extended versions of Kruglov et al. Mathematics Subject Classification 60F15
Highlights
1 Introduction The concept of complete convergence was introduced by Hsu and Robbins [1]
Hsu and Robbins [1] proved that the sequence of arithmetic means of i.i.d. random variables converges completely to the expected value if the variance of the summands is finite
Sung et al [3] obtained the following complete convergence theorem for arrays of rowwise independent random variables {Xni, 1 ≤ i ≤ kn, n ≥ 1}, where {kn, n ≥ 1 } is a sequence of positive integers
Summary
The concept of complete convergence was introduced by Hsu and Robbins [1]. A sequence {Xn, n ≥ 1} of random variables is said to converge completely to the constant θ if. Let {Xni, 1 ≤ i ≤ kn, n ≥ 1} be an array of rowwise negatively associated mean zero random variables and {an, n ≥ 1} a sequence of nonnegative constants. Chen et al [9] and Dehua et al [10] obtained complete convergence results (Theorems 1.4 and 1.6, respectively) for negatively associated and negatively dependent random variables and they proved the case of mean zero by using these results. Since e−x ≤ 1 ≤ 1 for x > 0, we get that for x > 0 and y > 0, 1+x x exp − x2
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