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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 ++X j+k , S n =X 1 ++X n , and Y k,j =f({S j (k)-kμ}/k), with μ 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,,n}, both in the case of k being a fixed positive integer as well as for k, but k/n0 as n. 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 M. Peligrad and Q.-M. Shao [ibid. 52, No. 1, 140-157 (1995; Zbl 0816.62027)] and M. Peligrad and R. Suresh [Stochastic Processes Appl. 56, No. 2, 307-319 (1995; Zbl 0817.62019)] in case of ρ-mixing variables, and by Zhang and Shi (1998) for negatively associated variables.

MSC:
60F15Strong limit theorems
60E15Inequalities in probability theory; stochastic orderings