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A note on the Kesten--Grincevičius--Goldie theorem (1512.07262v2)

Published 22 Dec 2015 in math.PR

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--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 $E |B|\kappa < \infty$. We assume that $E |B|\nu < \infty$ for some $\nu > \kappa$, and consider two cases (i) $E A\kappa = 1$, $E A\kappa \log_+ A = \infty$; (ii) $E A\kappa < 1$, $E At = \infty$ for all $t > \kappa$. 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.

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