## Electronic Communications in Probability

### Which distributions have the Matsumoto-Yor property?

#### Abstract

For four types of functions $\xi : ]0,\infty[\to ]0,\infty[$, we characterize the law of two independent and positive r.v.'s $X$ and $Y$ such that $U:=\xi(X+Y)$ and $V:=\xi(X)-\xi(X+Y)$ are independent. The case $\xi(x)=1/x$ has been treated by Letac and Wesolowski (2000). As for the three other cases, under the weak assumption that $X$ and $Y$ have density functions whose logarithm is locally integrable, we prove that the distribution of $(X,Y)$ is unique. This leads to Kummer, gamma and beta distributions. This improves the result obtained in [1] where more regularity was required from the densities.

#### Article information

Source
Electron. Commun. Probab., Volume 16 (2011), paper no. 49, 556-566.

Dates
Accepted: 29 September 2011
First available in Project Euclid: 7 June 2016

https://projecteuclid.org/euclid.ecp/1465262005

Digital Object Identifier
doi:10.1214/ECP.v16-1663

Mathematical Reviews number (MathSciNet)
MR2836761

Zentralblatt MATH identifier
1244.60016

Subjects
Primary: 60E05: Distributions: general theory
Secondary: 62E10: Characterization and structure theory

Rights
• Matsumoto, Hiroyuki; Yor, Marc. An analogue of Pitman's $2M-X$ theorem for exponential Wiener functionals. II. The role of the generalized inverse Gaussian laws. Nagoya Math. J. 162 (2001), 65–86.