Infinitely Divisible Distributions and Commutative Diagrams
Nomvelo Sibisi

TL;DR
This paper introduces the Lévy-Khintchine commutative diagram (LKCD), a visual tool for representing infinitely divisible distributions on the nonnegative real line, aiding in understanding and discovering new distributions.
Contribution
It develops the LKCD as a unified visual framework for ID distributions, especially gamma and related distributions, facilitating exploration and discovery of new cases.
Findings
Introduces the Lévy-Khintchine commutative diagram (LKCD) as a representational tool.
Shows how to generate new GGC distributions using convolutions and mixtures.
Provides LKCD representations for distributions involving special functions like Bessel and Mittag-Leffler.
Abstract
We study infinitely divisible (ID) distributions on the nonnegative half-line . 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 -stable distribution on for , which we regard as arising from the gamma distribution rather than as a separate object. It is characterised by its Laplace transform for . It is indeed often characterised…
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Taxonomy
TopicsMathematical functions and polynomials · Statistical Distribution Estimation and Applications · Statistical Mechanics and Entropy
