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Some limit theorems for the empirical process indexed by functions
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  • Published: March 1987

Some limit theorems for the empirical process indexed by functions

  • J. E. Yukich1 nAff2 

Probability Theory and Related Fields volume 74, pages 71–90 (1987)Cite this article

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  • 8 Citations

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Summary

Let

,n≧1, be a sequence of classes of real-valued measurable functions defined on a probability space (S,

,P). Under weak metric entropy conditions on

,n≧1, and under growth conditions on

we show that there are non-zero numerical constantsC 1 andC 2 such that

where α(n) is a non-decreasing function ofn related to the metric entropy of

. A few applications of this general result are considered: we obtain a.s. rates of uniform convergence for the empirical process indexed by intervals as well as a.s. rates of uniform convergence for the empirical characteristic function over expanding intervals.

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Author information

Author notes
  1. J. E. Yukich

    Present address: Department of Mathematics, Lehigh University, 18015, Bethlehem, PA, USA

Authors and Affiliations

  1. Département de Mathématiques, Université Louis Pasteur, F-67084, Strasbourg, France

    J. E. Yukich

Authors
  1. J. E. Yukich
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Additional information

Portions of this article were presented during the conference on Mathematical Stochastics (February 19–25, 1984) at Oberwolfach, West Germany

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Yukich, J.E. Some limit theorems for the empirical process indexed by functions. Probab. Th. Rel. Fields 74, 71–90 (1987). https://doi.org/10.1007/BF01845640

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  • Received: 27 June 1984

  • Revised: 07 April 1986

  • Issue Date: March 1987

  • DOI: https://doi.org/10.1007/BF01845640

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Keywords

  • Entropy
  • Growth Condition
  • Stochastic Process
  • Characteristic Function
  • Probability Theory
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