## Electronic Communications in Probability

### A note on the Kesten–Grincevičius–Goldie theorem

Péter Kevei

#### 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.

#### Article information

Source
Electron. Commun. Probab. Volume 21 (2016), paper no. 51, 12 pp.

Dates
Accepted: 15 July 2016
First available in Project Euclid: 26 July 2016

http://projecteuclid.org/euclid.ecp/1469557025

Digital Object Identifier
doi:10.1214/16-ECP9

Zentralblatt MATH identifier
1345.60021

#### Citation

Kevei, Péter. A note on the Kesten–Grincevičius–Goldie theorem. Electron. Commun. Probab. 21 (2016), paper no. 51, 12 pp. doi:10.1214/16-ECP9. http://projecteuclid.org/euclid.ecp/1469557025.

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