Subexponential Distributions

Abstract

Subexponential distributions are a special class of heavy‐tailed distributions. The name arises from one of their properties that their tails decrease slower than any exponential tail (3). This implies that large values can occur in a sample with nonnegligible probability, which makes the subexponential distributions natural candidates for modeling situations in which some extremely large values occur in a sample compared with the mean size of the data. Such a pattern is often seen in insurance data, for instance, in fire, wind–storm, or flood insurance (collectively known as catastrophe insurance). Subexponential claims can account for large fluctuations in the surplus process of a company, increasing the risk involved in such portfolios.

Asymptotics of discounted aggregate claims for renewal risk model with risky investment

“…In modelling extremal event, heavy-tailed risk has played an important role in insurance and finance, because it can describe large claims efficiently; see Embrechts et al [5], Kong [12,13] and Goldie & Klüppelberg [8] for nice reviews. We give here several important classes of heavy-tailed distributions for further references:…”

Large Insurance Losses Distributions

“…Various review papers have appeared on subexponentials; see e.g. [20,24,43]; textbook accounts are in [1,2,12,31,37,38]. We present two defining properties of subexponential d.f.s.…”