Renorming divergent perpetuities

Paweł Hitczenko; Jacek Wesołowski

doi:10.3150/10-BEJ297

paper

Renorming divergent perpetuities

arXiv:1107.2753 · doi:10.3150/10-BEJ297

Abstract

We consider a sequence of random variables $(R_{n})$ defined by the recurrence $R_{n} = Q_{n} + M_{n} R_{n - 1}$ , $n \geq 1$ , where $R_{0}$ is arbitrary and $(Q_{n}, M_{n})$ , $n \geq 1$ , are i.i.d. copies of a two-dimensional random vector $(Q, M)$ , and $(Q_{n}, M_{n})$ is independent of $R_{n - 1}$ . It is well known that if $E ln ∣ M ∣ < 0$ and $E ln^{+} ∣ Q ∣ < \infty$ , then the sequence $(R_{n})$ converges in distribution to a random variable $R$ given by $R = d \sum_{k = 1}^{\infty} Q_{k} \prod_{j = 1}^{k - 1} M_{j}$ , 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 $E ln ∣ M ∣$ 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.

Published in at http://dx.doi.org/10.3150/10-BEJ297 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)