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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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 $ \mathcal{F}$ of functions in the case of infinitely divisible purely Poisson (in particular, stable) Radon limits, under conditions on the local modulus of the processes $ \{ f(X):\,f \in \mathcal{F}\} $ ("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 $ {\mathbf{R}}$ and its associated empirical measures.


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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