The Class of Subexponential Distributions

Abstract

The class \\$\\mathscr{J}\\$ of subexponential distributions is characterized by \\$F(0) = 0, 1 - F^{(2)} (x) \\sim 2\\{1 - F(x)\\}\\$ as \\$x \\rightarrow \\infty\\$. New properties of the class \\$\\mathscr{J}\\$ are derived as well as for the more general case where \\$1 - F^{(2)} (x) \\sim \\beta\\{1 - F(x)\\}\\$. An application to transient renewal theory illustrates these results as does an adaptation of a result of Greenwood on randomly stopped sums of subexponentially distributed random variables.