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On Distributional Properties of Perpetuities
Authors:Gerold Alsmeyer  Alex Iksanov  Uwe Rösler
Institution:1. Institut für Mathematische Statistik, Westf?lische Wilhelms-Universit?t Münster, Einsteinstra?e 62, 48149, Münster, Germany
2. Faculty of Cybernetics, National T. Shevchenko University, 01033, Kiev, Ukraine
3. Mathematisches Seminar, Christian-Albrechts-Universit?t zu Kiel, Ludewig-Meyn-Str. 4, 24098, Kiel, Germany
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.
Keywords:Perpetuity  Continuity of distribution  Atom  Moment  Exponential moment  Abscissa of convergence
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