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.
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