On perpetuities with light tails

TL;DR

This paper investigates the asymptotic behavior of the logarithmic tails of perpetuities with light tails, revealing how dependence between components influences tail decay rates, contrasting with heavy-tailed cases.

Contribution

It provides precise asymptotics for tail probabilities of perpetuities with light tails under independence and bounds under dependence, highlighting the impact of dependence structures.

Findings

Asymptotic formulas for tail probabilities when M and Q are independent.

Asymptotic bounds for tail probabilities under dependence.

Dependence significantly affects tail decay rates, unlike in heavy-tailed cases.

Abstract

In the paper we consider the asymptotics of logarithmic tails of a perpetuity $$R \stackrel{d}{=}\sum_{j=1}^\infty Q_j \prod_{k=1}^{j-1}M_k,\qquad(M_n,Q_n)_{n=1}^\infty \mbox{ are i.i.d. copies of }(M,Q),$$ in the case when $\mathbb{P}(M\in[0,1))=1$ and $Q$ has all exponential moments. If $M$ and $Q$ are independent, under regular variation assumptions, we find the precise asymptotics of $-\log\mathbb{P}(R>x)$ as $x\to\infty$. Moreover, we deal with the case of dependent $M$ and $Q$ and give asymptotic bounds for $-\log\mathbb{P}(R>x)$. It turns out that dependence structure between $M$ and $Q$ has a significant impact on the asymptotic rate of logarithmic tails of $R$. Such phenomenon is not observed in the case of heavy-tailed perpetuities.