Renorming divergent perpetuities

TL;DR

This paper investigates the behavior of a recursive sequence of random variables, known as perpetuities, especially when the usual convergence conditions are not met, and explores whether normalized versions converge to a meaningful limit.

Contribution

It introduces a new perspective on perpetuities by analyzing their behavior under non-convergent conditions and proposes normalization techniques to study their limits.

Findings

Sequence $(R_n)$ may not converge without traditional conditions.

Normalized $(R_n)$ can converge to a non-degenerate limit.

Provides conditions for convergence under non-negative $E{ m ln}|M|$.

Abstract

We consider a sequence of random variables $(R_n)$ defined by the recurrence $R_n=Q_n+M_nR_{n-1}$, $n\ge1$, where $R_0$ is arbitrary and $(Q_n,M_n)$, $n\ge1$, 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 $R\stackrel{d}{=}\sum_{k=1}^{\infty}Q_k\prod_{j=1}^{k-1}M_j$, 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.