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

Abstract

In this paper we show under weak assumptions that for $R\stackrel{d}{=}1+M_1+M_1{M}_2+\dots$, where $P(M\in [0,1])=1$ and $M_i$ are independent copies of $M$, we have $\ln P(R>x)\sim Cx\ln P(M>1-\frac{1}{x})$ as $x\to \infty$. The constant $C$ is given explicitly and its value depends on the rate of convergence of $\ln P(M>1-\frac{1}{x})$. 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, of perpetuities.

21 pages

References in corpus (1)

• Two-sided bounds for $L_p$-norms of combinations of products of independent random variables

Cited by in corpus (3)

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