
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 in the case when and has all exponential moments. If and are independent, under regular variation assumptions, we find the precise asymptotics of as . Moreover, we deal with the case of dependent and and give asymptotic bounds for . It turns out that dependence structure between and has a significant impact on the asymptotic rate of logarithmic tails of . Such phenomenon is not observed in the case of heavy-tailed perpetuities.
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