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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 $X \stackrel{\mathcal {D}} {=} A X + B$, where $(A,B)$ and $X$ on the right-hand side are independent. The Kesten–Grincevičius–Goldie theorem states that if $\mathbf{E} A^\kappa = 1$, $\mathbf{E} A^\kappa \log _+ A < \infty $, and $\mathbf{E} |B|^\kappa < \infty $, then $\mathbf{P} \{ X > x \} \sim c x^{-\kappa }$. Assume that $\mathbf{E} |B|^\nu < \infty $ for some $\nu > \kappa $, and consider two cases (i) $\mathbf{E} A^\kappa = 1$, $\mathbf{E} A^\kappa \log _+ A = \infty $; (ii) $\mathbf{E} A^\kappa < 1$, $\mathbf{E} A^t = \infty $ for all $t > \kappa $. We show that under appropriate additional assumptions on $A$ the asymptotic $\mathbf{P} \{ X > x \} \sim c x^{-\kappa } \ell (x) $ holds, where $\ell $ 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

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