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The weak convergence for functions of negatively associated random variables. (English) Zbl 0989.60033
Consider a sequence of stationary, negatively associated random variables, and set $S_j(1)= X_{j+k}+\cdots+ X_{j+k}$, $S_n= X_1+\cdots+ X_n$, and $Y_{k,j}= f(\{S_j(k)- k\mu\}/\sqrt k)$, with $\mu$ denoting the mean of $X_1$, and $f$ a real function satisfying some conditions. The author’s main results provide an asymptotic Wiener process approximation for the partial sum process of the $\{Y_{k,j}: j=1,\dots, n\}$, both in the case of $k$ being a fixed positive integer as well as for $k\to \infty$, but $k/n\to 0$ as $n\to\infty$. A corresponding result for positively associated random variables is also presented. As a consequence of the main results, the author is able to derive asymptotic normality of some estimators of the asymptotic variance of $S_n$, which have earlier been discussed by {\it M. Peligrad} and {\it Q.-M. Shao} [ibid. 52, No. 1, 140-157 (1995; Zbl 0816.62027)] and {\it M. Peligrad} and {\it R. Suresh} [Stochastic Processes Appl. 56, No. 2, 307-319 (1995; Zbl 0817.62019)] in case of $\rho$-mixing variables, and by {\it Zhang} and {\it Shi} (1998) for negatively associated variables.

60F15Strong limit theorems
60E15Inequalities in probability theory; stochastic orderings
Full Text: DOI
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