Open Access
2016 A note on the Kesten–Grincevičius–Goldie theorem
Péter Kevei
Electron. Commun. Probab. 21: 1-12 (2016). DOI: 10.1214/16-ECP9

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}cxκ. 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}cxκ(x) holds, where is a nonconstant slowly varying function. We use Goldie’s renewal theoretic approach.

Citation

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Péter Kevei. "A note on the Kesten–Grincevičius–Goldie theorem." Electron. Commun. Probab. 21 1 - 12, 2016. https://doi.org/10.1214/16-ECP9

Information

Received: 21 January 2016; Accepted: 15 July 2016; Published: 2016
First available in Project Euclid: 26 July 2016

zbMATH: 1345.60021
Digital Object Identifier: 10.1214/16-ECP9

Subjects:
Primary: 60E99 , 60H25

Keywords: exponential functional , implicit renewal theorem , maximum of random walk , perpetuity equation , stochastic difference equation , strong renewal theorem

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