On the non-closure under convolution of the subexponential family

Abstract

A distribution F of a non-negative random variable belongs to the subexponential family of distributions S if 1 – F (2)(x) ~ 2(1 – F(x)) as x →∞. This family is of considerable interest in branching processes, queueing theory, transient renewal theory and infinite divisibility theory. Much is known about the kind of distributions that belong to S but the question of whether S is closed under convolution has remained unresolved for some time. This paper contains an example which demonstrates that S is not in fact closed.