Some new approaches to infinite divisibility

TL;DR

This paper explores new methods for establishing the infinite divisibility of certain probability distributions, building on classical results and introducing novel approaches based on polynomial zeros and Wiener-Hopf factorization.

Contribution

It introduces new results on infinite divisibility using approaches based on polynomial zeros and Wiener-Hopf factorization, extending classical theories.

Findings

Nondegenerate distributions with specific properties are infinitely divisible.

Discrete distributions that are log-convex or completely monotone are compound geometric and infinitely divisible.

New results enhance the understanding of infinite divisibility in probability theory.

Abstract

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.