The Annals of Applied Probability

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

Bartosz Kołodziejek

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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

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

Received: October 2015
Revised: May 2016
First available in Project Euclid: 26 May 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H25: Random operators and equations [See also 47B80]
Secondary: 60E99: None of the above, but in this section

Random iterated equation perpetuity regular variation tail asymptotic


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.

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