Electronic Communications in Probability

A note on the Kesten–Grincevičius–Goldie theorem

Péter Kevei

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1469557025

Digital Object Identifier
doi:10.1214/16-ECP9

Zentralblatt MATH identifier
1345.60021

Subjects
Primary: 60H25: Random operators and equations [See also 47B80] 60E99: None of the above, but in this section

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

Rights
Creative Commons Attribution 4.0 International License.

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


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