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April 2017 Logarithmic tails of sums of products of positive random variables bounded by one
Bartosz Kołodziejek
Ann. Appl. Probab. 27(2): 1171-1189 (April 2017). DOI: 10.1214/16-AAP1228

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.

Citation

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Bartosz Kołodziejek. "Logarithmic tails of sums of products of positive random variables bounded by one." Ann. Appl. Probab. 27 (2) 1171 - 1189, April 2017. https://doi.org/10.1214/16-AAP1228

Information

Received: 1 October 2015; Revised: 1 May 2016; Published: April 2017
First available in Project Euclid: 26 May 2017

zbMATH: 1370.60048
MathSciNet: MR3655863
Digital Object Identifier: 10.1214/16-AAP1228

Subjects:
Primary: 60H25
Secondary: 60E99

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.27 • No. 2 • April 2017
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