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Poisson and Gaussian approximation of weighted local empirical processes. (English) Zbl 0911.60004

Let \(S\) be a complete separable metric space with the Borel field \(\mathcal B .\) Let \(X_{n,i}\), \(i=1,\dots , n\), be independent identically distributed random elements with values in \((S,\mathcal B).\) Consider the empirical measure \(P_n(B) = \frac {1}{n}\sum _{i=1}^n \mathbb I _B (X_{n,i})\), \(B \in \mathcal B\), and the true probability measure \(P_{(n)}(B) = \mathbb P(X_{n,i} \in B),\;B\in \mathcal B. \) Poisson and Gaussian weighted approximations of the local empirical process defined by means of \(P_n(A) - P_{(n)}(A)\), \(A \in \mathcal A \subset \mathcal B\), are studied. It is shown that results for weighted local empirical processes indexed by a set obtained in the paper generalize many previous results on weighted empirical processes, that are now formulated as corollaries.

MSC:

60F05 Central limit and other weak theorems
62E20 Asymptotic distribution theory in statistics
62G30 Order statistics; empirical distribution functions
60F17 Functional limit theorems; invariance principles
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