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J Theor Probab DOI 10.1007/s10959-008-0156-8

Summary: J Theor Probab
DOI 10.1007/s10959-008-0156-8
On Distributional Properties of Perpetuities
Gerold Alsmeyer · Alex Iksanov · Uwe Rösler
Received: 15 October 2007 / Revised: 12 March 2008
© Springer Science+Business Media, LLC 2008
Abstract We study probability distributions of convergent random series of a special
structure, called perpetuities. By giving a new argument, we prove that such distrib-
utions 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 illus-
trated with a number of examples at the end of the article.
Keywords Perpetuity · Continuity of distribution · Atom · Moment · Exponential
moment · Abscissa of convergence
Mathematics Subject Classification (2000) Primary 60E99 · Secondary 60G50
G. Alsmeyer
Institut für Mathematische Statistik, Westfälische Wilhelms-Universität Münster, Einsteinstraße 62,
48149 Münster, Germany
e-mail: gerolda@math.uni-muenster.de


Source: Alsmeyer, Gerold - Institut für Mathematische Statistik, Westfälische Wilhelms-Universität Münster


Collections: Mathematics