## A note on the Kesten-Grincevičius-Goldie theorem.(English)Zbl 1345.60021

Summary: Consider the perpetuity equation $$X=AX+B$$ in distribution, where $$(A,B)$$ and $$X$$ on the right-hand side are independent. The Kesten-Grincevičius-Goldie theorem states that if $$\operatorname{E} A^{\kappa} = 1$$, $$\operatorname{E} A^{\kappa} \log _+ A < \infty$$ and $$\operatorname{E} |B|^{\kappa} < \infty$$, then $$\operatorname{P} \{ X > x \} \sim c x^{-\kappa}$$. Assume that $$\operatorname{E} |B|^{\nu} < \infty$$ for some $$\nu > \kappa$$, and consider two cases: (i) $$\operatorname{E} A^{\kappa} = 1$$, $$\operatorname{E} A^{\kappa} \log _+ A = \infty$$; (ii) $$\operatorname{E} A^{\kappa} < 1$$, $$\operatorname{E} A^t = \infty$$ for all $$t > \kappa$$. We show that under appropriate additional assumptions on $$A$$, the asymptotic result $$\operatorname{P} \{ X > x \} \sim c x^{-\kappa } \ell (x)$$ holds, where $$\ell$$ is a nonconstant slowly varying function. We use C. M. Goldie’s renewal theoretic approach [Ann. Appl. Probab. 1, No. 1, 126–166 (1991; Zbl 0724.60076)].

### MSC:

 60F10 Large deviations 60K05 Renewal theory 60G50 Sums of independent random variables; random walks 39A50 Stochastic difference equations 60H99 Stochastic analysis 60H25 Random operators and equations (aspects of stochastic analysis) 60E99 Distribution theory

Zbl 0724.60076
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