Identification of the residual term in multiplicative self-decomposition using Fox $H$-functions
José Luís da Silva, Mohamed Erraoui
TL;DR
The paper addresses the problem of identifying the residual term in multiplicative self-decompositions of key distributions. It employs a Mellin-transform framework to derive explicit residual densities: an $M$-Wright density for the exponential case and Fox $H$-function densities for the gamma and generalized gamma distributions, with equivalent Wright representations for related Gaussian-absolute-value structures. The main contributions are explicit residual pdfs, Mellin-transform-based characterizations that uniquely identify the original distributions under multiplicative self-decomposition, and a unified Fox $H$-function representation that encompasses the GG family. This work provides a rigorous analytic foundation for multiplicative self-decomposability, enabling exact residual modeling for products of independent variables and connections to self-similar processes.
Abstract
Multiplicative self-decomposable laws describe random variables that can be decomposed into a product of a scaled-down version of themselves and an independent residual term. Shanbhag et al.~(1977) have shown that the gamma distribution is multiplicative self-decomposable, in particular, the exponential distribution. 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 paper, we aim to fill this gap by providing 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.