Logarithmic tails of sums of products of positive random variables bounded by one

Abstract

In this paper, we show under weak assumptions that for $R\stackrel{d}{=}1+M_{1}+M_{1}M_{2}+\cdots$, where $\mathbb{P}(M\in[0,1])=1$ and $M_{i}$ are independent copies of $M$, we have $\ln\mathbb{P}(R>x)\sim Cx\ln\mathbb{P}(M>1-1/x)$ as $x\to\infty$. The constant $C$ is given explicitly and its value depends on the rate of convergence of $\ln\mathbb{P}(M>1-1/x)$. Random variable $R$ satisfies the stochastic equation $R\stackrel{d}{=}1+MR$ with $M$ and $R$ independent, thus this result fits into the study of tails of iterated random equations, or more specifically, perpetuities.