On perpetuities with light tails

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 ass...

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Bibliographic Details
Published inAdvances in applied probability Vol. 50; no. 4; pp. 1119 - 1154
Main Author Kołodziejek, Bartosz
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.12.2018
Applied Probability Trust
Subjects
Asymptotic properties
Dependence
Original Article
Probability
convex conjugate
Tauberian theorem
Primary 60H25
Secondary 60E99
dependence structure
Perpetuity
regular variation
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Summary: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.
Bibliography:ObjectType-Article-1
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content type line 14
ISSN:0001-8678
1475-6064
DOI:10.1017/apr.2018.53