A note on the Kesten--Grincevičius--Goldie theorem

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 $P \{ X > x \} \sim c x^{-κ}$ if $E A^κ= 1$, $E A^κ\log_+ A < \infty$, and $E |B|^κ< \infty$. We assume that $E |B|^ν< \infty$ for some $ν> κ$, and consider two cases (i) $E A^κ= 1$, $E A^κ\log_+ A = \infty$; (ii) $E A^κ< 1$, $E A^t = \infty$ for all $t > κ$. We show that under appropriate additional assumptions on $A$ the asymptotic $P \{ X > x \} \sim c x^{-κ} \ell(x) $ holds, where $\ell$ is a nonconstant slowly varying function. We use Goldie's renewal theoretic approach.