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

TL;DR

This paper explicitly identifies the distribution of the residual term in multiplicative self-decomposable laws, using Fox $H$-functions, extending previous results by providing concrete distribution forms for specific cases.

Contribution

It derives explicit Fox $H$-function based distributions for residual terms in multiplicative self-decomposability, filling a gap in prior theoretical work.

Findings

Residual term for exponential distribution follows an $M$-Wright distribution.

Residual distribution for generalized gamma and normal variables is an $H_{1,1}^{1,0}$ Fox $H$-distribution.

Provides explicit formulas for residual distributions in multiplicative self-decomposition.

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.