Convergence rate for randomly weighted sums of random variables and its application

Let $\{X,X_n,\, n\geq1\}$ be a sequence of identically distributed negatively superadditive-dependent random variables, and let $\{A_{ni},\, 1\leq i\leq n,\, n \!\geq\! 1\}$ be an array of negatively supperadditive-dependent random weights. Under the almost optimal moment conditions, we show that for any $\varepsilon>0$, $\sum_{n=1}^{\infty}n^{-1}\mathbf{P}\bigl(\max_{1\leq m\leq n} \bigl| \sum_{i=1}^mA_{ni}X_i\bigr|>\varepsilon n^{1/\alpha}\ln^{1/\gamma}n\bigr) <\infty$, where $0<\gamma<\alpha\leq2$, and that for any $0<q<\alpha$, $\sum_{n=1}^{\infty}n^{-1}\mathbf E\bigl(n^{-1/\alpha}\ln^{-1/\gamma}n\max_{1\leq m\leq n} \bigl| \sum_{i=1}^mA_{ni}X_i\bigr|-\varepsilon\bigr)_+^q<\infty$. The main results obtained here extend and improve the corresponding ones in the literature. As an application, a new result on the strong law of large numbers for the random weighting estimation of sample mean is provided.