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Extreme value theory for a class of discrete distributions with applications to some stochastic processes

Published online by Cambridge University Press:  14 July 2016

C. W. Anderson
C. W. Anderson*
Affiliation:
Imperial College, London
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Abstract

Let ξn be the maximum of a set of n independent random variables with common distribution function F whose support consists of all sufficiently large positive integers. Some of the classical asymptotic results of extreme value theory fail to apply to ξn for such F and this paper attempts to find weaker ones which give some description of the behaviour of ξn as n → ∞. These are then applied to the extreme value theory of certain regenerative stochastic processes.

Type
Research Papers
Information
Journal of Applied Probability , Volume 7 , Issue 1 , April 1970 , pp. 99 - 113
Copyright
Copyright © Applied Probability Trust 

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References

[1] Barndorff-Nielsen, O. (1964) On the limit distribution of the maximum of a random number of independent random variables. Act. Math. Acad. Sci. Hung. 15, 399–403.CrossRefGoogle Scholar
[2] Berman, S. M. (1962) Limiting distribution of the maximum term in sequences of dependent random variables. Ann. Math. Statist. 33, 894–908.Google Scholar
[3] Cohen, J. W. (1967) The distribution of the maximum number of customers present simultaneously during a busy period for the queueing systems M/G/1 and GI/M/1. J. Appl. Prob. 4, 162–179.Google Scholar
[4] Feller, W. (1966) An Introduction to Probability Theory and Its Applications , Vol. 2. Wiley, New York.Google Scholar
[5] Gnedenko, B. V. (1943) Sur la distribution limite du terme maximum d'une série aléatoire. Ann. Math. 44, 423–453.CrossRefGoogle Scholar
[6] Lukacs, E. (1960) Characteristic Functions. Griffin, London.Google Scholar
[7] Smirnov, N. V. (1952) Limit distributions for the terms of a variational series. Amer. Math. Soc. Transl. No. 67.Google Scholar
[8] Smith, W.L. (1958) Renewal theory and its ramifications. J.R. Statist. Soc. B 20, 243–302.Google Scholar