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 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.