Advances in Applied Probability

On fixed points of Poisson shot noise transforms

Aleksander M. Iksanov and Zbigniew J. Jurek

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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.

Article information

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

Dates
First available in Project Euclid: 22 November 2002

Permanent link to this document
http://projecteuclid.org/euclid.aap/1037990954

Digital Object Identifier
doi:10.1239/aap/1037990954

Mathematical Reviews number (MathSciNet)
MR1938943

Zentralblatt MATH identifier
1030.60011

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

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

Citation

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. http://projecteuclid.org/euclid.aap/1037990954.


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