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Monatshefte für Mathematik

, Volume 153, Issue 3, pp 183–204 | Cite as

Entropy conditions for subsequences of random variables with applications to empirical processes

  • István Berkes
  • Walter Philipp
  • Robert Tichy
Article

Abstract.

We introduce new entropy concepts measuring the size of a given class of increasing sequences of positive integers. Under the assumption that the entropy function of \({\cal A}\) is not too large, many strong limit theorems will continue to hold uniformly over all sequences in \({\cal A}\). We demonstrate this fact by extending the Chung-Smirnov law of the iterated logarithm on empirical distribution functions for independent identically distributed random variables as well as for stationary strongly mixing sequences to hold uniformly over all sequences in \({\cal A}\). We prove a similar result for sequences (n k ω) mod 1 where the sequence (n k ) of real numbers satisfies a Hadamard gap condition.

2000 Mathematics Subject Classification: 11K38, 60F15 
Key words: Discrepancy, entropy, random numbers 

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • István Berkes
    • 1
  • Walter Philipp
    • 2
  • Robert Tichy
    • 1
  1. 1.Technical University GrazAustria
  2. 2.University of IllinoisChampaignUSA

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