Advances in Applied Probability

On fixed points of Poisson shot noise transforms

Aleksander M. Iksanov and Zbigniew J. Jurek

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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.

Article information

Adv. in Appl. Probab. Volume 34, Number 4 (2002), 798-825.

First available in Project Euclid: 22 November 2002

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E07: Infinitely divisible distributions; stable distributions
Secondary: 60K05: Renewal theory

Shot noise transform fixed points regular variation renewal theorem absolute continuity infinite divisibility Banach contraction principle


Iksanov, Aleksander M.; Jurek, Zbigniew J. On fixed points of Poisson shot noise transforms. Adv. in Appl. Probab. 34 (2002), no. 4, 798--825. doi:10.1239/aap/1037990954.

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