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On a stochastic difference equation and a representation of non–negative infinitely divisible random variables

Published online by Cambridge University Press:  01 July 2016

Wim Vervaat
Wim Vervaat*
Affiliation:
Katholieke Universiteit, Nijmegen
*
∗Postal address: Mathematisch Instituut, Katholieke Universiteit, Toernooiveld, Nijmegen, The Netherlands.
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Abstract

The present paper considers the stochastic difference equation Yn = AnYn-1 + Bn with i.i.d. random pairs (An, Bn) and obtains conditions under which Yn converges in distribution. This convergence is related to the existence of solutions of and (A, B) independent, and the convergence w.p. 1 of ∑ A1A2 ··· An-1Bn. A second subject is the series ∑ Cnf(Tn) with (Cn) a sequence of i.i.d. random variables, (Tn) the sequence of points of a Poisson process and f a Borel function on (0, ∞). The resulting random variable turns out to be infinitely divisible, and its Lévy–Hinčin representation is obtained. The two subjects coincide in case An and Cn are independent, Bn = AnCn, An = U1/αn with Un a uniform random variable, f(x) = e−x/α.

Keywords

STOCHASTIC DIFFERENCE EQUATIONSHOT NOISEINFINITE DIVISIBILITYGROUP OF AFFINE TRANSFORMATIONSRANDOM WALK IN A GROUPHARMONIC FUNCTIONPOISSON PROCESSLÉVY–HINČIN REPRESENTATIONCENTRAL LIMIT THEOREMSTABLE DISTRIBUTIONSSELF-DECOMPOSABLE DISTRIBUTIONS
Type
Research Article
Information
Advances in Applied Probability , Volume 11 , Issue 4 , December 1979 , pp. 750 - 783
Copyright
Copyright © Applied Probability Trust 1979 

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Footnotes

Research supported in part by the Netherlands Organization for the Advancement of Pure Research (ZWO).

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