Infinitely Divisible Distributions and Commutative Diagrams

Abstract

We study infinitely divisible (ID) distributions on the nonnegative half-line $\mathbb{R}_+$. The L\'{e}vy-Khintchine representation of such distributions is well-known. Our primary contribution is to cast the probabilistic objects and the relations amongst them in a unified visual form that we refer to as the L\'{e}vy-Khintchine commutative diagram (LKCD). While it is introduced as a representational tool, the LKCD facilitates the exploration of new ID distributions and may thus also be looked upon, at least in part, as a discovery tool. The basic object of the study is the gamma distribution. Closely allied to this is the $\alpha$-stable distribution on $\mathbb{R}_+$ for $0<\alpha<1$, which we regard as arising from the gamma distribution rather than as a separate object. It is characterised by its Laplace transform $\exp(-s^\alpha)$ for $0<\alpha<1$. It is indeed often characterised as an instance of a class of ID distributions known as generalised gamma convolutions (GGCs). We make use of convolutions and mixtures of gamma and stable densities to generate densities of other GGC distributions, with particular cases involving Bessel, confluent hypergeometric, Mittag-Leffler and parabolic cylinder functions. We present all instances as LKCD representations.