Electronic Communications in Probability

Which distributions have the Matsumoto-Yor property?

Angelo Koudou and Pierre Vallois

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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

Permanent link to this document
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

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

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

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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References

  • E. Koudou and P. Vallois. Independence properties of the Matsumoto-Yor type. To appear in Bernoulli.
  • G. Letac. The random continued fractions of Dyson and their extension. Talk at Charles University, Prague, November 25, 2009.
  • Letac, Gérard; Wesołowski, Jacek. An independence property for the product of GIG and gamma laws. Ann. Probab. 28 (2000), no. 3, 1371–1383.
  • 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.
  • Wesołowski, Jacek. On a functional equation related to the Matsumoto-Yor property. Aequationes Math. 63 (2002), no. 3, 245–250.