Complete convergence and almost sure convergence of weighted sums of random variables (Q1345079)

The main aim of this paper is to give a general treatment of complete convergence of weighted sums of independent random variables (i.r.v.'s). A general results on complete convergence under some very relaxed conditions is \[ \sum^ \infty_{n=1} n^{r-2} P \left \{\Bigl | \sum^ \infty_{i= -\infty} X_{ni} \Bigr | \geq \varepsilon \right\} < \infty \quad \text{for every }\varepsilon > 0, \] where \(\mu>1\), \((X_{ni}\), \(-\infty < i < \infty)\) is a sequence of i.r.v.'s for each \(n \geq 1\). This result is used in the study of complete and almost sure (a.s.) convergence of weighted sums of i.r.v.'s. We give only the next interesting result: Let \((X,X_ n,\;n \geq 1)\) be a sequence of i.i.d. r.v.'s with \(E | X |^ p < \infty\) for some \(p \geq 1\) and \(EX = 0\). Let \((i_ n,\;n \geq 1)\) be a sequence of positive integers and \((a_{ni};\;1 \leq i \leq i_ n\), \(n \geq 1)\) an array of real numbers satisfying: (1) \(i_ n = O(n)\); (2) \(\sup_{n,i} | a_{ni} | < \infty\); (3) \(\sum^{i_ n}_{i=1} a^ 2_{ni} = O(n^ \delta)\) for some \(\delta < \min \{1,2/p\}\). Then \[ \lim_{n \to \infty}n^{-1/p} \sum^{i_ n}_{i=1} a_{ni} X_ i = 0 \text{ a.s.}. \]