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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 Y n = A n Y n-1 + B n with i.i.d. random pairs (A n , B n ) and obtains conditions under which Y n converges in distribution. This convergence is related to the existence of solutions of and (A, B) independent, and the convergence w.p. 1 of ∑ A 1 A 2 ··· A n-1 B n . A second subject is the series ∑ C n f(T n ) with (C n ) a sequence of i.i.d. random variables, (T n ) 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 A n and C n are independent, B n = A n C n , A n = U 1/α n with U n 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

Information

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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