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On fixed points of Poisson shot noise transforms

Part of: Distribution theory - Probability Special processes

Published online by Cambridge University Press:  01 July 2016

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

Distributional fixed points of a Poisson shot noise transform (for nonnegative and nonincreasing response functions bounded by 1) are characterized. The tail behavior of fixed points is described. Typically they have either exponential moments or their tails are proportional to a power function, with exponent greater than −1. The uniqueness of fixed points is also discussed. Finally, it is proved that in most cases fixed points are absolutely continuous, apart from the possible atom at zero.

Keywords

Shot noise transformfixed pointsregular variationrenewal theoremabsolute continuityinfinite divisibilityBanach contraction principle

MSC classification

Primary: 60E07: Infinitely divisible distributions; stable distributions
Secondary: 60K05: Renewal theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 34 , Issue 4 , December 2002 , pp. 798 - 825
Copyright
Copyright © Applied Probability Trust 2002 

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Footnotes

∗∗

Current address: Department of Mathematics, Wayne State University, Detroit, MI 48202, USA.

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