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Uniform convergence of sums of order statistics to stable laws
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  • Published: June 1988

Uniform convergence of sums of order statistics to stable laws

  • A. Janssen1 

Probability Theory and Related Fields volume 78, pages 261–271 (1988)Cite this article

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Summary

Let X 1, X 2,... denote an i.i.d. sequence of real valued random variables which ly in the domain of attraction of a stable law Q with index 0<α<1. Under a von Mises condition we show that the sum of order statistics

$$a_n^{ - 1} \left( {\sum\limits_{i = 1}^{k(n)} {X_{i:n} + \sum\limits_{i = n + 1 - r(n)}^n {X_{i:n} } } } \right)$$

converges to Q with respect to the norm of total variation if for instance min(k(n), r(n))→∞.

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Authors and Affiliations

  1. Fachbereich Mathematik, Universitäts-GH-Siegen, Hölderlinstrasse 3, D-5900, Siegen, Federal Republic of Germany

    A. Janssen

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  1. A. Janssen
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Janssen, A. Uniform convergence of sums of order statistics to stable laws. Probab. Th. Rel. Fields 78, 261–271 (1988). https://doi.org/10.1007/BF00322023

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  • Received: 05 May 1987

  • Revised: 12 November 1987

  • Issue Date: June 1988

  • DOI: https://doi.org/10.1007/BF00322023

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Keywords

  • Total Variation
  • Stochastic Process
  • Probability Theory
  • Statistical Theory
  • Order Statistic
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