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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
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