On perpetuities with light tails

TL;DR

This work analyzes the logarithmic tails of a light-tailed perpetuity $R$ generated by $(M,Q)$ with ${\rm P}(M\in[0,1))=1$ and exponentially light $Q$. Employing regular variation and Tauberian theory, it introduces a variational tail scale $h(x)=\inf_{t\ge1}\{-t\log {\mathbb P}((1)/(1-M)>t,Q> x/t)\}$ and proves sharp tail asymptotics in the independent case, with $-\log {\mathbb P}(R>x)\sim c\,h(x)$ where $c$ depends on the regular variation indices; for bounded $Q$ a related form in terms of $f(x/q_+)$ appears. In the dependent case, the paper provides asymptotic bounds that demonstrate how the dependence between $M$ and $Q$ can significantly impact the logarithmic tail rate, unlike in heavy-tailed settings. The results are built on a Tauberian framework (Kasahara/de Bruijn) and yield a unified view of tail behavior across independence and dependence, with precise variational characterizations and extremal bounds via comonotonic/countermonotonic constructions.

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.

Theorems & Definitions (19)

• Example 1.1
• Lemma 2.1
• Theorem 2.1
• Theorem 2.2
• Lemma 2.2
• Remark 2.1
• Theorem 2.3: Kasahara's Tauberian Theorem
• Theorem 2.4: de Bruijn's Tauberian Theorem
• Theorem 3.1
• Theorem 3.2