On the non-closure under convolution for strong subexponential distributions

In this paper, we consider the convolution closure problem for the class of strong subexponential distributions, denoted as S*. First, we show that, if F, G ∈ L, then inclusions of F*G, FG, and pF + (1 – p)G for all (some) p ∈ (0; 1) into the class S* are equivalent. Then, using examples constructed by Klüppelberg and Villasenor [The full solution of the convolution closure problem for convolution-equivalent distributions, J. Math. Anal. Appl., 41:79–92, 1991], we show that S* is not closed under convolution.