Almost sure convergence for weighted sums of extended negatively dependent random variables (Q343210)

Let \(\{a_{n,k}, 1\leq k \leq n,\, n\geq 1 \}\) be a triangular array of real numbers, and let \(\{X_n, n \geq 1 \}\) be a sequence of random variables. Under suitable conditions on the sequence \(\{X_n, n\geq 1 \}\), the author shows that \(\sum_{k=1}^{n} a_{n,k} (X_k-\operatorname{E}X_k) \buildrel{a.s.}\over\longrightarrow 0\). The underlying conditions are that the sequence \(\{X_n,n \geq 1\}\) under consideration is extended negatively dependent and either (i) stochastically dominated by a random variable \(X\) such that \(\operatorname{E}| X| ^p < \infty \) for some \(1<p<2\) or (ii) the sequence \(\{X_n, n \geq 1\}\) is identically distributed such that \(\operatorname{E}| X_1| <\infty \).