×

zbMATH — the first resource for mathematics

Normal and stable convergence of integral functions of the empirical distribution function. (English) Zbl 0589.60030
Different kinds of the invariance principle say that the uniform empirical \(\{\alpha_ n(s)\); \(0\leq s\leq 1\}\) resp. quantile process \(\{u_ n(s)\); \(0\leq s\leq 1\}\) can be approximated (in some sense) by a sequence of Brownian bridges \(\{B_ n(s)\); \(0\leq s\leq 1\}\). Let f(s) be a weight function on (0,1) with \(\lim_{s\to 0}f(s)=\lim_{s\to 1}f(s)=\infty.\) The main aim of the authors is to prove that the sequence \(\{B_ n(s)\); \(0\leq s\leq 1\}\) can be defined in such a way that \(f(s)| B_ n(s)-\alpha_ n(s)|\) resp. \(f(s)| B_ n(s)-u_ n(s)|\) should be small nearly over the whole interval (0,1).
They also investigate how an integral \(\int g(s)du_ n(s)\) can be approximated by \(\int g(s)dB_ n(s)\) where g(s) belongs to some set of real functions satisfying some regularity conditions. An analogous problem is to approximate the integral \(\int \alpha_ n(s)dQ(s)\) by the integral \(\int B_ n(s)dQ(s)\) where Q(\(\cdot)\) is the inverse of a given distribution function.
Reviewer: P.Révész

MSC:
60F17 Functional limit theorems; invariance principles
62G30 Order statistics; empirical distribution functions
60F05 Central limit and other weak theorems
60E07 Infinitely divisible distributions; stable distributions
PDF BibTeX XML Cite
Full Text: DOI