ON THE WEAK LAW FOR WEIGHTED SUMS INDEXED BY RANDOM VARIABLES UNDER NEGATIVELY ASSOCIATED ARRAYS

• Published : 2003.01.01
• 56 5

Abstract

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

Keywords

negatively associated random variables;weighted sums indexed by random variables;weak law of large numbers;martingale difference sequence

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