A note on the Kesten--Grincevi\v{c}ius--Goldie theorem

Consider the perpetuity equation $XAX+B$, where $(A,B)$ and $X$ on the right-hand side are independent. The Kesten–Grincevi\v{c}ius–Goldie theorem states that $P { X > x } \sim c x^{-\kappa}$if$E A^\kappa = 1$,$E A^\kappa \log_+ A < \infty$,and$E |B|^\kappa < \infty$\mathrm{.}Weassumethat$E |B|^\nu < \infty$forsome$\nu > \kappa$,andconsidertwocases(i)$E A^\kappa = 1$,$E A^\kappa \log_+ A = \infty$;(ii)$E A^\kappa < 1$,$E A^t = \infty$forall$t > \kappa$. We show that under appropriate additional assumptions on $A$ the asymptotic $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.