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# On a Stochastic Difference Equation and a Representation of Non-Negative Infinitely Divisible Random Variables

Wim Vervaat
Vol. 11, No. 4 (Dec., 1979), pp. 750-783
DOI: 10.2307/1426858
Stable URL: http://www.jstor.org/stable/1426858
Page Count: 34
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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 $Y\overset \text{d}\to{=}AY+B,\ Y$ 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=Un 1/α with Un a uniform random variable, f(x)=e-x/α.

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