<abstract><p>The authors study the convergence rate of complete moment convergence for weighted sums of weakly dependent random variables without assumptions of identical distribution. Under the moment condition of $ E{{{\left| X \right|}^{\alpha }}}/{{{\left(\log \left(1+\left| X \right| \right) \right)}^{\alpha /\gamma -1}}}\; &lt; \infty $ for $ 0 &lt; \gamma &lt; \alpha $ with $ 1 &lt; \alpha \le 2 $, we establish the complete $ \alpha $-th moment convergence theorem for weighted sums of weakly dependent cases, which improves and extends the related known results in the literature.</p></abstract>

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In this work, suppose that {X n ; n ≥ 1}is a sequence of asymptotically negatively associated random variables and {a ni ; 1 ≤ i ≤ n, n ≥ 1} is an array of real numbers such that ∑ i = 1 n | a n i | q = O ( n ) for some q > max { α p − 1 α − 1 / 2 , 2 } with αp > 1 and α > 1 2 . Let l (x) > 0 be a slowly varying function at infinity. We establish some equivalent conditions of the complete convergence for weighted sums of this form ∑ n = 1 ∞ n α p − 2 l ( n ) P ( max 1 ≤ j ≤ n | ∑ i = 1 j a n i X i | > ε n α ) < ∞   for   all   ε   >   0 . As applications, some strong laws of large numbers for weighted sums of asymptotically negatively associated random variables are also obtained.