On the weak law for weighted sums indexed by random variables under negatively associated arrays

Abstract

Let $\{ X_{n k}~ | 1 \leq k \le n, n \ge 1 \}$ be an array of row negatively associated ($NA$) random variables which satisfy $P(|X_{nk} | >x) \le P(|X|>x)$. For weighed sums $ \sum_{k=1}^{T_n} a_k X_{nk}$ indexed by random variables $\{T_n | n \geq 1 \}$, we establish a general weak law of large numbers ($WLLN$) of the form $ \left( \sum_{k=1}^{T_n} a_k X_{nk} - \nu_{[nk]} \right) / b_{[\alpha_n]}$ under some suitable conditions, where $\{a_n | n \ge 1 \}$, $\{b_n | n \ge 1 \}$ are sequences of constants with $a_n > 0,~ 0<b_n \rightarrow \infty,~ n \ge 1$, and $\{ \nu_{\alpha_n} | n \ge 1 \}$ is an array of random variables, and the symbol [$x$] denotes the greatest integer in $x$.