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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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