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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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