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On perpetuities with light tails

Part of: Distribution theory - Probability Stochastic analysis

Published online by Cambridge University Press:  29 November 2018

Bartosz Kołodziejek
Bartosz Kołodziejek*
Affiliation:
Warsaw University of Technology
*
* Postal address: Faculty of Mathematics and Information Science, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland. Email address: b.kolodziejek@mini.pw.edu.pl
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Abstract

In this paper we consider the asymptotics of logarithmic tails of a perpetuity R=D∑j=1∞Qj∏k=1j-1Mk, where (Mn,Qn)n=1∞ are independent and identically distributed copies of (M,Q), for the case when ℙ(M∈[0,1))=1 and Q has all exponential moments. If M and Q are independent, under regular variation assumptions, we find the precise asymptotics of -logℙ(R>x) as x→∞. Moreover, we deal with the case of dependent M and Q, and give asymptotic bounds for -logℙ(R>x). It turns out that the dependence structure between M and Q has a significant impact on the asymptotic rate of logarithmic tails of R. Such a phenomenon is not observed in the case of heavy-tailed perpetuities.

Keywords

Perpetuitydependence structureregular variationTauberian theoremconvex conjugate

MSC classification

Primary: 60H25: Random operators and equations
Secondary: 60E99: None of the above, but in this section
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 4 , December 2018 , pp. 1119 - 1154
Copyright
Copyright © Applied Probability Trust 2018 

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