A look at perpetuities via asymptotically homogeneous in space Markov chains

TL;DR

This paper explores the behavior of perpetuities using asymptotically homogeneous Markov chains, deriving new tail asymptotics and limit theorems, and improving existing moment condition results in the literature.

Contribution

It introduces a Markov chain framework for analyzing perpetuities, providing new asymptotic tail results and enhanced moment conditions for stable and unstable cases.

Findings

New tail asymptotics for stable perpetuities

Limit theorems for unstable perpetuities

Improved moment conditions in tail analysis

Abstract

It is shown how a natural representation of perpetuities as asymptotically homogeneous in space Markov chains allows to prove various asymptotic tail results for stable perpetuities and limit theorems for unstable ones. Some of these results are new while others essentially improve moment conditions known in the literature. Both subexponential and Cram\'er's cases are considered.