Electronic Communications in Probability

Which distributions have the Matsumoto-Yor property?

Angelo Koudou and Pierre Vallois

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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

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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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Gamma distribution generalized inverse Gaussian distribution Matsumoto-Yor property Kummer distribution Beta distribution

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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

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