Independence properties of the Matsumoto--Yor type

TL;DR

This paper characterizes specific decreasing functions that relate to independence properties of certain random variables, identifying four main functions under additional assumptions and deriving related distributional convolution results.

Contribution

The paper classifies all LWMY functions satisfying the independence property, revealing four main functions and providing new convolution relations involving gamma and Kummer distributions.

Findings

Identified four main LWMY functions under additional assumptions

Established that $f(x)=1/x$ corresponds to the Matsumoto-Yor property

Derived new convolution relations involving gamma and Kummer distributions

Abstract

We define Letac-Wesolowski-Matsumoto-Yor (LWMY) functions as decreasing functions from $(0,\infty)$ onto $(0,\infty)$ with the following property: there exist independent, positive random variables $X$ and $Y$ such that the variables $f(X+Y)$ and $f(X)-f(X+Y)$ are independent. We prove that, under additional assumptions, there are essentially four such functions. The first one is $f(x)=1/x$. In this case, referred to in the literature as the Matsumoto-Yor property, the law of $X$ is generalized inverse Gaussian while $Y$ is gamma distributed. In the three other cases, the associated densities are provided. As a consequence, we obtain a new relation of convolution involving gamma distributions and Kummer distributions of type 2.