A property of longtailed distributions. (English) Zbl 0534.60015

Embrechts, Paul; Omey, Edward

J. Appl. Probab. 21, 80-87 (1984).

Various classes of longtailed distributions and their interrelationship are studied. the important class of subexponential distributions, i.e. those F for which \(1-F^{(2)}(x)\to 2(1-F(x))\) as \(x\to \infty\), where \(F^{(2)}(x)\) denotes the convolution of F with itself, is studied more in detail. Some necessary conditions are given so that the integrated tail distribution \(F_ 1(x)=m^{-1}\int^{x}_{0}(1-F(y))dy\) of a general F is subexponential. These results are applied to yield a heavytailed alternative to the classical Cramér-Lundberg estimate in ruin theory, as well as to prove some second-order rate of convergence results in the elementary renewal theorem.

MSC:

60E05	Probability distributions: general theory
60K05	Renewal theory

Keywords:

integrated tail distribution; risk theory; subexponentiality