## The Annals of Applied Probability

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

Bartosz Kołodziejek

#### 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.

#### Article information

Source
Ann. Appl. Probab., Volume 27, Number 2 (2017), 1171-1189.

Dates
Revised: May 2016
First available in Project Euclid: 26 May 2017

https://projecteuclid.org/euclid.aoap/1495764376

Digital Object Identifier
doi:10.1214/16-AAP1228

Mathematical Reviews number (MathSciNet)
MR3655863

Zentralblatt MATH identifier
1370.60048

Subjects
Secondary: 60E99: None of the above, but in this section

#### Citation

Kołodziejek, Bartosz. Logarithmic tails of sums of products of positive random variables bounded by one. Ann. Appl. Probab. 27 (2017), no. 2, 1171--1189. doi:10.1214/16-AAP1228. https://projecteuclid.org/euclid.aoap/1495764376

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