  Using an approach based, amongst other things, on Proposition 1 of Kaluza
(1928), Goldie (1967) and, using a different approach based especially on zeros
of polynomials, Steutel (1967) have proved that each nondegenerate distribution
function (d.f.) $F$ (on $\RR$, the real line), satisfying $F(0-) = 0$ and $F(x)
= F(0) + (1-F(0)) G(x)$, $x > 0$, where $G$ is the d.f. corresponding to a
mixture of exponential distributions, is infinitely divisible. Indeed,
Proposition 1 of Kaluza (1928) implies that any nondegenerate discrete
probability distribution ${p_x: x= 0,1, ...}$ that is log-convex or, in
particular, completely monotone, is compound geometric, and, hence, infinitely
divisible. Steutel (1970), Shanbhag & Sreehari (1977) and Steutel & van Harn
(2004, Chapter VI) have given certain extensions or variations of one or more
of these results. Following a modified version of the C.R. Rao et al. (2009,
Section 4) approach based on the Wiener-Hopf factorization, we establish some
further results of significance to the literature on infinite divisibility.
