On the Laplace transform of perpetuities with thin tails

TL;DR

This paper investigates the precise bounds of the Laplace transform for solutions to a specific distributional equation, revealing detailed tail behavior of these perpetuities with thin tails.

Contribution

It provides exact lower and upper bounds for the Laplace transform domain of perpetuities satisfying a linear distributional equation.

Findings

Determines the precise domain of the Laplace transform for these perpetuities.

Establishes bounds that characterize the tail decay of solutions.

Extends understanding of tail behavior beyond exponential bounds.

Abstract

We consider the random variables $R$ which are solutions of the distributional equation $R\overset{\cL}{=}MR+Q$, where $(Q,M)$ is independent of $R$ and $\ABS{M}\leq 1$. Goldie and Gr\"ubel showed that the tails of $R$ are no heavier than exponential. In this note we provide the exact lower and upper bounds of the domain of the Laplace transform of $R$.