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Infinite divisibility of infinite sums of lower records: a simple proof

Part of: Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Aleksander M. Iksanov
Aleksander M. Iksanov*
Affiliation:
Taras Shevchenko National University of Kiev
*
∗ Postal address: Faculty of Cybernetics, Taras Shevchenko National University of Kiev, 01033 Kiev, Ukraine. Email address: iksan@unicyb.kiev.ua
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Abstract

Let {X n , n = 0, 1,…} be the sequence of the lower records for an arbitrary underlying distribution μ on [0, ∞). We show that is equal in distribution to where {τ i , i = 1, 2,…} is a Poisson flow of unit intensity and g is a right-continuous and nonincreasing function defined by μ. This observation allows us to extend results of Bose et al. and simplify their proofs.

Keywords

Lower recordinfinite divisibilityshot noise distribution

MSC classification

Primary: 60E07: Infinitely divisible distributions; stable distributions 60E10: Characteristic functions; other transforms
Type
Short Communications
Information
Journal of Applied Probability , Volume 41 , Issue 4 , December 2004 , pp. 1187 - 1190
Copyright
Copyright © Applied Probability Trust 2004 

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References

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