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Renorming divergent perpetuities
Journal article   Open access   Peer reviewed

Renorming divergent perpetuities

Pawel Hitczenko and Jacek Wesolowski
Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability, v 17(3), pp 880-894
01 Aug 2011
DOI: https://doi.org/10.3150/10-BEJ297
url
https://doi.org/10.3150/10-bej297View
Published, Version of Record (VoR)Open Access (License Unspecified),  Open
url
https://doi.org/10.3150/10-BEJ297View
Published, Version of Record (VoR) Open

Abstract

Mathematics Physical Sciences Science & Technology Statistics & Probability
We consider a sequence of random variables (R-n) defined by the recurrence R-n = Q(n) + MnRn-1, n >= 1, where Ro is arbitrary and (Q(n), M-n), n >= 1 , are i.i.d. copies of a two-dimensional random vector (Q, M), and (Q(n), M-n) is independent of Rn-1. It is well known that if Eln vertical bar M vertical bar <0 and Eln(+) vertical bar Q vertical bar < infinity, then the sequence (R-n) converges in distribution to a random variable R given by R (d) double under bar Sigma(infinity)(k=1) Q(k)Pi(k-1)(j=1), and usually referred to as perpetuity. In this paper we consider a situation in which the sequence (R-n) itself does not converge. We assume that Eln vertical bar M vertical bar exists but that it is non-negative and we ask if in this situation the sequence (R-n), after suitable normalization, converges in distribution to a non-degenerate limit.

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