Some characterizations of multiple selfdecomposability with extensions and an application to the Gamma function (2103.10160v3)

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.