On fixed points of Poisson shot noise transforms

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, wit...

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Bibliographic Details
Published in:Advances in applied probability 2002-12, Vol.34 (4), p.798-825
Main Authors: Iksanov, Aleksander M., Jurek, Zbigniew J.
Format: Article
Language:English
Subjects:
60E07
60K05
absolute continuity
Banach contraction principle
Eigenfunctions
Fixed point property
fixed points
Fourier transformations
General Applied Probability
Hubble Space Telescope
infinite divisibility
Mathematical models
Mathematical theorems
Probability
Probability distributions
Random variables
regular variation
renewal theorem
Shot noise
Shot noise transform
Studies
Urelements
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Summary: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.
ISSN:0001-8678
1475-6064
DOI:10.1239/aap/1037990954