The Annals of Applied Probability

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

Bartosz Kołodziejek

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

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

Keywords
Random iterated equation perpetuity regular variation tail asymptotic

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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