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New Explicit Examples of Fixed Points of Poisson Shot Noise Transforms

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  • Aleksander M. Iksanov
  • Che Soong Kim

Abstract

The main objective of this paper is to establish a close relation between the fixed points of Poisson shot noise transforms and perpetuities of a special type. With this relation it is shown that the gamma distributions, the generalized positive Linnik distributions, and the S2 distributions are fixed points of Poisson shot noise transforms. The paper also proves that log‐convexity of the response functions is not needed for non‐negative Poisson shot noise distributions to be self‐decomposable. Finally, the problems of existence and uniqueness of the above mentioned perpetuities are investigated.

Suggested Citation

  • Aleksander M. Iksanov & Che Soong Kim, 2004. "New Explicit Examples of Fixed Points of Poisson Shot Noise Transforms," Australian & New Zealand Journal of Statistics, Australian Statistical Publishing Association Inc., vol. 46(2), pages 313-321, June.
    • Handle: RePEc:bla:anzsta:v:46:y:2004:i:2:p:313-321
    DOI: 10.1111/j.1467-842X.2004.00332.x
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    Cited by:

    1. Iksanov, Aleksander M., 2004. "Elementary fixed points of the BRW smoothing transforms with infinite number of summands," Stochastic Processes and their Applications, Elsevier, vol. 114(1), pages 27-50, November.
    2. Iksanov, A.M.Aleksander M. & Kim, Che-Soong, 2004. "On a Pitman-Yor problem," Statistics & Probability Letters, Elsevier, vol. 68(1), pages 61-72, June.
    3. Gerold Alsmeyer & Alex Iksanov & Uwe Rösler, 2009. "On Distributional Properties of Perpetuities," Journal of Theoretical Probability, Springer, vol. 22(3), pages 666-682, September.

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