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The structure of the class of subexponential distributions
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  • Published: December 1988

The structure of the class of subexponential distributions

  • Eric Willekens1 

Probability Theory and Related Fields volume 77, pages 567–581 (1988)Cite this article

Summary

LetX 1,X 2, ...,X n be a sequence of positive, independent, identically distributed random variables with the same distribution function (d.f.)F and denote byX 1:n ≦X 2:n ≦...≦X n:n the order statistics of the sample. We characterize the class of d.f.F for which

$$P(X_{1:n} + X_{2:n} + \ldots + X_{n - i:n} > x) \sim P(X_{n - i:n} > x) as x \to \infty $$

for fixedn andi (i≦n-1), and we show that it is independent ofn. This leads to the genesis of a new class of d.f.L i ; we show that the sequence (L i ) ∞ i =0 is strictly decreasing and we illustrate how the classesL i determine the probabilistic structure of the classL of subexponential distributions.

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Authors and Affiliations

  1. Departement Wiskunde, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, B-3030, Heverlee, Belgium

    Eric Willekens

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  1. Eric Willekens
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Research Assistent of the Belgian National Fund for Scientific Research

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Willekens, E. The structure of the class of subexponential distributions. Probab. Th. Rel. Fields 77, 567–581 (1988). https://doi.org/10.1007/BF00959618

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  • Received: 22 August 1986

  • Issue Date: December 1988

  • DOI: https://doi.org/10.1007/BF00959618

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Keywords

  • Distribution Function
  • Stochastic Process
  • Probability Theory
  • Order Statistic
  • Mathematical Biology
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