Complete convergence and strong laws of large numbers for weighted sums of negatively orthant dependent random variables (Q519944)

Let \(\{X_{n},n\geq 1\}\) be a sequence of identically distributed negatively orthant dependent random variables. Let \(\{a_{ni},1\leq i\leq n,n\geq 1\}\) be an array of constants such that \(\sum_{i=1}^{n}\left| a_{ni}\right| ^{\alpha}=O(n),\) for some \(0<\alpha <2.\) Let us put \( b_{n}=n^{1/\alpha}(\log n)^{1/\gamma},\) for some \(\gamma >\alpha .\) The main result of this paper presents necessary and sufficient conditions under which, for every \(\varepsilon >0,\) \(\sum_{n=1}^{\infty}n^{-1}P(\max_{1\leq m\leq n}\) \(\left| \sum_{i=1}^{m}a_{ni}X_{i}\right| >\varepsilon b_{n})<\infty .\) As a consequence of this result, a strong law of large numbers for weighted sums of negatively orthant dependent random variables is presented, too.