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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The central limit theorem for empirical processes under local conditions: the case of Radon infinitely divisible limits without Gaussian component
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by Niels T. Andersen, Evarist Giné and Joel Zinn PDF
Trans. Amer. Math. Soc. 308 (1988), 603-635 Request permission

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.
References
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Additional Information
  • © Copyright 1988 American Mathematical Society
  • 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