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Title:

A note on the Kesten-Grincevičius-Goldie theorem

Document type:
Zeitschriftenaufsatz
Author(s):
Kevei, P.
Abstract:
Consider the perpetuity equation X=DAX+B, where (A,B) and X on the right-hand side are independent. The Kesten–Grincevičius–Goldie theorem states that if EAκ=1, EAκlog+A<∞, and E|B|κ<∞, then P{X>x}∼cx−κ. Assume that E|B|ν<∞ for some ν>κ, and consider two cases (i) EAκ=1, EAκlog+A=∞; (ii) EAκ<1, EAt=∞ for all t>κ. We show that under appropriate additional assumptions on A the asymptotic P{X>x}∼cx−κℓ(x) holds, where ℓ is a nonconstant slowly varying function. We use Goldie’s renewal theoretic approach.
Keywords:
perpetuity equation; stochastic difference equation; strong renewal theorem; expo- nential functional; maximum of random walk; implicit renewal theorem
Journal title:
Electronic Communications in Probability
Year:
2016
Journal volume:
21
Year / month:
2016-07
Quarter:
3. Quartal
Month:
Jul
Journal issue:
51
Pages contribution:
1-12
Reviewed:
nein
Language:
en
Fulltext / DOI:
doi:10.1214/16-ECP9
Status:
Erstveröffentlichung
TUM Institution:
Lehrstuhl für Mathematische Statistik
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