The Relations And Applications Of Some Heavy-tailed Distribution Classes

Since the 1960s, heavy-tailed distributions, especially subexponential distributions,have been widely used in random walk, branching process, queueing theory, risk theoryand other fields. Among the heavy-tailed distributions, Lognormal distribution andWeibull distribution are one of the most typical distributions in insurance. In this paper,we discuss the relations and applications of some heavy-tailed distribution classes. InChapter 1, we introduce some notions, classifications and basic properties of heavy-taileddistributions. In Chapter 2, we discuss some equivalent conditions of the relationsbetween S~* and S_â–³, which are the most important subclasses of subexponential distri-bution class S. Particularly, subexponential behaviour and local subexponential behave-our of Lognormal distribution, Weibull distribution and their relational distributions areinvestigated. In Chapter 3, using the distribution theory, we discuss the precise largedeviations of heavy-tailed random sums and the integral representation of the tailasymptopics of the maximum on a random time interval of a random walk with heavytailed increments.