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On the maximum and its uniqueness for geometric random samples

Published online by Cambridge University Press:  14 July 2016

F. Thomas Bruss  and
Colm Art O'cinneide
F. Thomas Bruss
Affiliation:
University of Arizona
Colm Art O'cinneide*
Affiliation:
University of Arkansas
*
∗∗Postal address: Department of Mathematical Sciences, SE 301, University of Arkansas, Fayetteville, AR 72701, USA.
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Abstract

Given n independent, identically distributed random variables, let ρ n denote the probability that the maximum is unique. This probability is clearly unity if the distribution of the random variables is continuous. We explore the asymptotic behavior of the ρ n's in the case of geometric random variables. We find a function Φsuch that (ρ n – Φ(n)) → 0 as n →∞. In particular, we show that ρ n does not converge as n →∞. We derive a related asymptotic result for the expected value of the maximum of the sample. These results arose out of a random depletion model due to Bajaj, which was the original motivation for this paper and which is included.

Keywords

DEPLETION MODELCOIN TOSSINGORDER STATISTICSGENERATING FUNCTIONSTIES FOR THE MAXIMUM
Type
Research Papers
Information
Journal of Applied Probability , Volume 27 , Issue 3 , September 1990 , pp. 598 - 610
Copyright
Copyright © Applied Probability Trust 1990 

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Footnotes

∗

Present address: Department of Mathematics, UCLA, 405 Hilgard Avenue, Los Angeles, CA 90024-1555, USA.

Supported by the Alexander-van-Humboldt Foundation.

Supported under a full-time research assignment from the University of Arkansas.

References

Bahadur, R. R. (1971) Some Limit Theorems in Statistics. CBMS Regional Conference Series in Applied Mathematics, 4.Google Scholar
David, H. A. (1981) Order Statistics. Wiley, New York.Google Scholar
Hardy, G. H. (1967) Divergent Series. Oxford University Press, London.Google Scholar
Stein, C. (1986) Approximate Computation of Expectations. IMS Lecture Notes – Monograph Series 7.Google Scholar