Bernoulli

  • Bernoulli
  • Volume 9, Number 3 (2003), 467-496.

Regular variation in the mean and stable limits for Poisson shot noise

Claudia Klüppelberg, Thomas Mikosch, and Anette Schärf

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Abstract

Poisson shot noise is a natural generalization of a compound Poisson process when the summands are stochastic processes starting at the points of the underlying Poisson process. We study the limiting behaviour of Poisson shot noise when the limits are infinite-variance stable processes. In this context a sufficient condition for this convergence turns up which is closely related to multivariate regular variation -- we call it regular variation in the mean. We also show that the latter condition is necessary and sufficient for the weak convergence of the point processes constructed from the normalized noise sequence and also for the weak convergence of its extremes.

Article information

Source
Bernoulli, Volume 9, Number 3 (2003), 467-496.

Dates
First available in Project Euclid: 6 October 2003

Permanent link to this document
https://projecteuclid.org/euclid.bj/1065444814

Digital Object Identifier
doi:10.3150/bj/1065444814

Mathematical Reviews number (MathSciNet)
MR1997493

Zentralblatt MATH identifier
1044.60013

Keywords
extremes infinitely divisible \rm dst Poisson random measure self-similar process stable process weak convergence

Citation

Klüppelberg, Claudia; Mikosch, Thomas; Schärf, Anette. Regular variation in the mean and stable limits for Poisson shot noise. Bernoulli 9 (2003), no. 3, 467--496. doi:10.3150/bj/1065444814. https://projecteuclid.org/euclid.bj/1065444814


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