Probability Theory and Related Fields

, Volume 77, Issue 4, pp 567–581 | Cite as

The structure of the class of subexponential distributions

  • Eric Willekens


LetX1,X2, ...,X n be a sequence of positive, independent, identically distributed random variables with the same distribution function (d.f.)F and denote byX1:nX2:n≦...≦Xn: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.


Distribution Function Stochastic Process Probability Theory Order Statistic Mathematical Biology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Eric Willekens
    • 1
  1. 1.Departement WiskundeKatholieke Universiteit LeuvenHeverleeBelgium

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