The central limit theorem for empirical processes under local conditions: the case of Radon infinitely divisible limits without Gaussian component

Authors:
Niels T. Andersen, Evarist Giné and Joel Zinn

Journal:
Trans. Amer. Math. Soc. **308** (1988), 603-635

MSC:
Primary 60F17; Secondary 60F05

DOI:
https://doi.org/10.1090/S0002-9947-1988-0930076-3

MathSciNet review:
930076

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Abstract: Weak convergence results are obtained for empirical processes indexed by classes of functions in the case of infinitely divisible purely Poisson (in particular, stable) Radon limits, under conditions on the local modulus of the processes ("bracketing" conditions). They extend (and slightly improve upon) a central limit theorem of Marcus and Pisier (1984) for Lipschitzian processes. The law of the iterated logarithm is also considered. The examples include Marcinkiewicz type laws of large numbers for weighted empirical processes and for the dual-bounded-Lipschitz distance between a probability in and its associated empirical measures.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1988-0930076-3

Keywords:
Central limit theorems,
law of the iterated logarithm,
empirical processes,
Marcinkiewicz laws of large numbers,
bracketing conditions,
majorizing measures

Article copyright:
© Copyright 1988
American Mathematical Society