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

TL;DR

This paper analyzes the asymptotic behavior of the tail probabilities of a perpetuity-like random variable R, showing that under weak conditions, its logarithmic tail is proportional to x times the logarithm of the tail of M near 1.

Contribution

It provides a precise asymptotic relation for the tail of R in terms of the tail of M, with an explicit constant depending on the convergence rate.

Findings

Derived the asymptotic tail behavior of R as x→∞

Established the relation between tail of R and tail of M near 1

Provided explicit constant C depending on convergence rate

Abstract

In this paper we show under weak assumptions that for $R\stackrel{d}{=}1+M_1+M_1M_2+\ldots$, where $P(M\in[0,1])=1$ and $M_i$ are independent copies of $M$, we have $\ln P(R>x)\sim C\, x\ln P(M>1-\frac1x)$ as $x\to\infty$. The constant $C$ is given explicitly and its value depends on the rate of convergence of $\ln P(M>1-\frac1x)$. 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, of perpetuities.