Identification of the residual term in multiplicative self-decomposition using Fox $H$-functions

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.