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Open Access
August 2011 Renorming divergent perpetuities
Paweł Hitczenko, Jacek Wesołowski
Bernoulli 17(3): 880-894 (August 2011). DOI: 10.3150/10-BEJ297

Abstract

We consider a sequence of random variables (Rn) defined by the recurrence Rn=Qn+MnRn1,n1, where R0 is arbitrary and (Qn,Mn),n1, are i.i.d. copies of a two-dimensional random vector (Q,M), and (Qn,Mn) is independent of Rn1. It is well known that if Eln|M|<0 and Eln+|Q|<, then the sequence (Rn) converges in distribution to a random variable R given by Rd=k=1Qkk1j=1Mj, and usually referred to as perpetuity. In this paper we consider a situation in which the sequence (Rn) itself does not converge. We assume that Eln|M| exists but that it is non-negative and we ask if in this situation the sequence (Rn), after suitable normalization, converges in distribution to a non-degenerate limit.

Citation

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Paweł Hitczenko. Jacek Wesołowski. "Renorming divergent perpetuities." Bernoulli 17 (3) 880 - 894, August 2011. https://doi.org/10.3150/10-BEJ297

Information

Published: August 2011
First available in Project Euclid: 7 July 2011

zbMATH: 1232.60017
MathSciNet: MR2817609
Digital Object Identifier: 10.3150/10-BEJ297

Keywords: Convergence in distribution , perpetuity , stochastic difference equation

Rights: Copyright © 2011 Bernoulli Society for Mathematical Statistics and Probability

Vol.17 • No. 3 • August 2011
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