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Weak convergence for empirical processes of associated sequences. (English) Zbl 0968.60019
Let $$(X_n)_{n\in Z}$$ be a stationary and associated sequence of random variables with common continuous distribution function $$F$$. For every integer $$n\geq 1$$ let $$F_n$$ denote the empirical distribution function based on $$X_1,\dots,X_n$$, and $$G_n=\sqrt{n}(F_n-F)$$ the corresponding empirical process. Let $$G$$ be the centered Gaussian process with covariance function $$\text{cov}(G(x),G(y))=\sum_{k\in Z}\text{cov} (1_{X_0\leq x},1_{X_k\leq y})$$. It is shown that the condition $$\text{cov}(F(X_1),F(X_n))=O(n^{-b})$$ for some $$b>4$$ implies weak convergence of $$G_n$$ to $$G$$ in the Skorokhod space $$D[-\infty,\infty]$$. This improves earlier results in which the condition on $$b$$ was more stringent. Applications to linear processes are also given, as well as a comparison of mixing conditions and association.

##### MSC:
 60F05 Central limit and other weak theorems 60F17 Functional limit theorems; invariance principles 62G30 Order statistics; empirical distribution functions
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