Poisson and Gaussian approximation of weighted local empirical processes
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.
scientific article; zbMATH DE number 1240040
scientific article