Subexponential Distributions

Abstract

As we stated in the Introduction, all those heavy-tailed distributions likely to be of use in practical applications are not only long-tailed but possess the additional regularity property of subexponentiality. Essentially this corresponds to good tail behaviour under the operation of convolution. In this chapter, following established tradition, we introduce first subexponential distributions on the positive half-line ${\mathbb{R}}^{+}$. It is not immediately obvious from the definition, but it nevertheless turns out, that subexponentiality is a tail property of a distribution. It is thus both natural, and important for many applications, to extend the concept to distributions on the entire real line $\mathbb{R}$. We also study the very useful subclass of subexponential distributions which was originally called ${\mathcal{S}}^{{_\ast}}$ in [29] and which we name strong subexponential. In particular this class again contains all those heavy-tailed distributions likely to be encountered in practice.