# zbMATH — the first resource for mathematics

A property of longtailed distributions. (English) Zbl 0534.60015
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
Full Text: