*(English)*Zbl 0989.60033

Consider a sequence of stationary, negatively associated random variables, and set ${S}_{j}\left(1\right)={X}_{j+k}+\cdots +{X}_{j+k}$, ${S}_{n}={X}_{1}+\cdots +{X}_{n}$, and ${Y}_{k,j}=f(\{{S}_{j}\left(k\right)-k\mu \}/\sqrt{k})$, with $\mu $ denoting the mean of ${X}_{1}$, and $f$ a real function satisfying some conditions. The authorâ€™s main results provide an asymptotic Wiener process approximation for the partial sum process of the $\{{Y}_{k,j}:j=1,\cdots ,n\}$, 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 *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 $\rho $-mixing variables, and by *Zhang* and *Shi* (1998) for negatively associated variables.