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A clustering law for some discrete order statistics

Published online by Cambridge University Press:  14 July 2016

Sunder Sethuraman*
Affiliation:
Iowa State University
*
Postal address: 400 Carver Hall, Department of Mathematics, Iowa State University, Ames, IA 50011, USA. Email address: sethuram@iastate.edu

Abstract

Let X1, X2, …, Xn be a sequence of independent, identically distributed positive integer random variables with distribution function F. Anderson (1970) proved a variant of the law of large numbers by showing that the sample maximum moves asymptotically on two values if and only if F satisfies a ‘clustering’ condition, In this article, we generalize Anderson's result and show that it is robust by proving that, for any r ≥ 0, the sample maximum and other extremes asymptotically cluster on r + 2 values if and only if Together with previous work which considered other asymptotic properties of these sample extremes, a more detailed asymptotic clustering structure for discrete order statistics is presented.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 2003 

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Footnotes

Research supported in part by NSF grant DMS-00711504.

References

Anderson, C. W. (1970). Extreme value theory for a class of discrete distributions with applications. J. Appl. Prob. 7, 99113.CrossRefGoogle Scholar
Athreya, J. S., and Sethuraman, S. (2001). On the asymptotics of discrete order statistics. Statist. Prob. Lett. 54, 243249.Google Scholar
Baryshnikov, Y., Eisenberg, B., and Stengle, G. (1995). A necessary and sufficient condition for the existence of the limiting probability of a tie for first place. Statist. Prob. Lett. 23, 203209.Google Scholar
Galambos, J. (1987). The Asymptotic Theory of Extreme Order Statistics. Krieger, Melbourne, FL.Google Scholar
Gnedenko, B. V. (1943). Sur la distribution limite du terme maximum d'une série aléatoire. Ann. Math. 44, 423453.Google Scholar