Electronic Communications in Probability
- Electron. Commun. Probab.
- Volume 16 (2011), paper no. 49, 556-566.
Which distributions have the Matsumoto-Yor property?
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  where more regularity was required from the densities.
Electron. Commun. Probab., Volume 16 (2011), paper no. 49, 556-566.
Accepted: 29 September 2011
First available in Project Euclid: 7 June 2016
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Koudou, Angelo; Vallois, Pierre. Which distributions have the Matsumoto-Yor property?. Electron. Commun. Probab. 16 (2011), paper no. 49, 556--566. doi:10.1214/ECP.v16-1663. https://projecteuclid.org/euclid.ecp/1465262005