×

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

Citations:

Zbl 0724.60076
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid