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On Distributional Properties of Perpetuities

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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.

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Authors and Affiliations

  1. Institut für Mathematische Statistik, Westfälische Wilhelms-Universität Münster, Einsteinstraße 62, 48149, Münster, Germany

    Gerold Alsmeyer

  2. Faculty of Cybernetics, National T. Shevchenko University, 01033, Kiev, Ukraine

    Alex Iksanov

  3. Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, 24098, Kiel, Germany

    Uwe Rösler

Authors
  1. Gerold Alsmeyer
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  2. Alex Iksanov
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  3. Uwe Rösler
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Correspondence to Alex Iksanov.

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Alsmeyer, G., Iksanov, A. & Rösler, U. On Distributional Properties of Perpetuities. J Theor Probab 22, 666–682 (2009). https://doi.org/10.1007/s10959-008-0156-8

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  • DOI: https://doi.org/10.1007/s10959-008-0156-8

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