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On Distributional Properties of Perpetuities

Author

Listed:
  • Gerold Alsmeyer

    (Westfälische Wilhelms-Universität Münster)

  • Alex Iksanov

    (National T. Shevchenko University)

  • Uwe Rösler

    (Christian-Albrechts-Universität zu Kiel)

Abstract

We study probability distributions of convergent random series of a special structure, called perpetuities. By giving a new argument, we prove that such distributions are of pure type: degenerate, absolutely continuous, or continuously singular. We further provide necessary and sufficient criteria for the finiteness of p-moments, p>0, as well as exponential moments. In particular, a formula for the abscissa of convergence of the moment generating function is provided. The results are illustrated with a number of examples at the end of the article.

Suggested Citation

  • Gerold Alsmeyer & Alex Iksanov & Uwe Rösler, 2009. "On Distributional Properties of Perpetuities," Journal of Theoretical Probability, Springer, vol. 22(3), pages 666-682, September.
    • Handle: RePEc:spr:jotpro:v:22:y:2009:i:3:d:10.1007_s10959-008-0156-8
    DOI: 10.1007/s10959-008-0156-8
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    References listed on IDEAS

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    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. 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.
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    Cited by:

    1. Jaakko Lehtomaa, 2015. "Asymptotic Behaviour of Ruin Probabilities in a General Discrete Risk Model Using Moment Indices," Journal of Theoretical Probability, Springer, vol. 28(4), pages 1380-1405, December.
    2. Gerold Alsmeyer & Alexander Iksanov & Matthias Meiners, 2015. "Power and Exponential Moments of the Number of Visits and Related Quantities for Perturbed Random Walks," Journal of Theoretical Probability, Springer, vol. 28(1), pages 1-40, March.
    3. Anita Behme & Alexander Lindner, 2015. "On Exponential Functionals of Lévy Processes," Journal of Theoretical Probability, Springer, vol. 28(2), pages 681-720, June.

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