Perpetuities with thin tails revisited

TL;DR

This paper investigates the tail behavior of solutions to a distributional equation involving random variables, focusing on how the behavior of the multiplicative component near 1 influences tail heaviness, especially in the case of constant additive term.

Contribution

It extends previous work by analyzing the connection between the tail behavior of solutions and the behavior of the multiplicative term near 1, specifically for constant Q and nonnegative M.

Findings

Tails are no heavier than exponential.

When Q is bounded and M resembles a uniform distribution near 1, tails are Poissonian.

Focus on the case with constant Q and nonnegative M.

Abstract

We consider the tail behavior of random variables $R$ which are solutions of the distributional equation $R\stackrel{d}{=}Q+MR$, where $(Q,M)$ is independent of $R$ and $|M|\le 1$. Goldie and Gr\"{u}bel showed that the tails of $R$ are no heavier than exponential and that if $Q$ is bounded and $M$ resembles near 1 the uniform distribution, then the tails of $R$ are Poissonian. In this paper, we further investigate the connection between the tails of $R$ and the behavior of $M$ near 1. We focus on the special case when $Q$ is constant and $M$ is nonnegative.