Michna, Zbigniew Self-similar processes in collective risk theory. (English) Zbl 0937.60029 J. Appl. Math. Stochastic Anal. 11, No. 4, 429-448 (1998). Conditions are given for an insurance risk process to converge weakly to a long-range dependent process, in particular fractional Brownian motion. Convergence of the associated ruin probabilities is discussed and various estimates of the ruin probabilities for fractional Brownian motion are given. The occurrence of long-range dependence in insurance risk is motivated by an example with alternating periods of safe and risky conditions, along the lines of D. Heath, S. Resnick and G. Samorodnitsky [Ann. Appl. Probab. 9, No. 2, 352-375 (1999)]. Reviewer: S.Asmussen (Lund) Cited in 16 Documents MSC: 60G15 Gaussian processes 60G70 Extreme value theory; extremal stochastic processes 91B30 Risk theory, insurance (MSC2010) Keywords:alternating renewal process; fractional Brownian motion; functional central limit theorem; Hermite rank; long-range dependence; ruin probability; Skorokhod topology PDFBibTeX XMLCite \textit{Z. Michna}, J. Appl. Math. Stochastic Anal. 11, No. 4, 429--448 (1998; Zbl 0937.60029) Full Text: DOI EuDML