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Central limit theorems for stochastic processes under random entropy conditions
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  • Published: July 1987

Central limit theorems for stochastic processes under random entropy conditions

  • Kenneth S. Alexander1 nAff2 

Probability Theory and Related Fields volume 75, pages 351–378 (1987)Cite this article

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Summary

Necessary and sufficient conditions are found for the weak convergence of the row sums of an infinitesimal row-independent triangular array (φ nj ) of stochastic processes, indexed by a set S, to a sample-continuous Gaussian process, when the array satisfies a “random entropy” condition, analogous to one used by Giné and Zinn (1984) for empirical processes. This entropy condition is satisfied when S is a class of sets or functions with the Vapnik-Ĉervonenkis property and each φ nj (f)fdνnj is of the form νnjc for some reasonable random finite signed measure v nj. As a result we obtain necessary and sufficient conditions for the weak convergence of (possibly non-i.i.d.) partial-sum processes, and new sufficient conditions for empirical processes, indexed by Vapnik-Ĉervonenkis classes. Special cases include Prokhorov's (1956) central limit theorem for empirical processes, and Shorack's (1979) theorems on weighted empirical processes.

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Author information

Author notes
  1. Kenneth S. Alexander

    Present address: Department of Mathematics, University of Southern California, 90089, Los Angeles, CA, USA

Authors and Affiliations

  1. Department of Statistics GN22, University of Washington, 98195, Seattle, WA, USA

    Kenneth S. Alexander

Authors
  1. Kenneth S. Alexander
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Additional information

Research supported by an NSF Postdoctoral Fellowship, grant no. MCS83-111686

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Alexander, K.S. Central limit theorems for stochastic processes under random entropy conditions. Probab. Th. Rel. Fields 75, 351–378 (1987). https://doi.org/10.1007/BF00318707

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  • Received: 27 February 1985

  • Issue Date: July 1987

  • DOI: https://doi.org/10.1007/BF00318707

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Keywords

  • Entropy
  • Stochastic Process
  • Probability Theory
  • Limit Theorem
  • Statistical Theory
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