Identification of the residual term in multiplicative self-decomposition using Fox $H$-functions
A talk in the Bielefeld Stochastic Afternoon series by
José Luís da Silva
| Abstract: | Multiplicative self-decomposable laws describe random variables that can be decomposed into a product of a scaled-down version of the random variable and an independent residual term. Shanbhag et al.~(1977) showed that the gamma distribution is multiplicative self-decomposable; in particular, the exponential distribution is. As a result, they established the multiplicative self-decomposability of the absolute value of a centered normal random variable. A limitation of Shanbhag's result is that the distribution of the residual component is not explicitly
identified. In this talk, we provide an
explicit distribution of the residual term using a Fox $H$-function. More precisely, the residual term follows an $M$-Wright distribution for the exponential distribution, whereas for the generalized gamma distribution and the absolute value of a centered normal random variable, an $H_{1,1}^{1,0}$ distribution with different parameters. This talk is based on a preliminary version \cite{Silva2025a} of the results presented in this presentation. J.~L. da~Silva and M.~Erraoui. \newblock Multiplicative self-decomposition of the exponential and gamma distributions, pp.~1-25, \newblock \href{https://arxiv.org/abs/2507.23467v1}{arXiv:2507.23467v1}, 2025. Within the CRC this talk is associated to the project(s): A5, B1 |