On perpetuities with light tails

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.

33 pages

References in corpus (2)

• On perpetuities with gamma-like tails
• Stochastic recursions: between Kesten's and Grincevičius-Grey's assumptions

Cited by in corpus (1)

• Stochastic recursions: between Kesten's and Grincevičius-Grey's assumptions