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

2021/03/18 by Jedidi, Wissem, Jurek, Zbigniew J., Romian, Jumanah Al
#60E05 #60E07 #60E10 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2103.10160

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 \mathbbG_t and log \mathbbG_t, t>0 variables are selfdecomposable. In fact, we show that log \mathbbG_t is twice selfdecomposable if, and only if, t≥ t₁ ≈ 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.

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