Some characterizations of multiple selfdecomposability with extensions and an application to the Gamma function

TL;DR

This paper explores multiple selfdecomposability properties of gamma-related distributions, introduces new factorizations of the Gamma function, and proposes tools for characterizing multiply selfdecomposable distributions using differential operators and stochastic integrals.

Contribution

It provides new characterizations of multiple selfdecomposability, extends the class of such distributions, and offers novel factorizations of the Gamma function and distributions.

Findings

Log Gamma variables are twice selfdecomposable if t ≥ 0.15165.

New factorizations of the Gamma function and distributions are established.

Tools based on Mellin-Euler operators are introduced for class characterization.

Abstract

Inspirations for this paper can be traced to Urbanik (1972) where convolution semigroups of multiple decomposable distributions were introduced. In particular, the classical gamma $\mathbb{G}_t$ and $\log \mathbb{G}_t$, $t>0$ variables are selfdecomposable. In fact, we show that $\log \mathbb{G}_t$ is twice selfdecomposable if, and only if, $t\geq t_1 \approx 0.15165$. Moreover, we provide several new factorizations of the Gamma function and the Gamma distributions. To this end, we revisit the class of multiply selfdecomposable distributions, denoted $L_n(R)$, and propose handy tools for its characterization, mainly based on the Mellin-Euler's differential operator. Furthermore, we also give a perspective of generalization of the class $L_n(R)$ based on linear operators or on stochastic integral representations.