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Kummer and gamma laws through independences on trees—Another parallel with the Matsumoto–Yor property

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  • Piliszek, Agnieszka
  • Wesołowski, Jacek

Abstract

The paper develops a rather unexpected parallel to the multivariate Matsumoto–Yor (MY) property on trees considered in Massam and Wesołowski (2004). The parallel concerns a multivariate version of the Kummer distribution, which is generated by a tree. Given a tree of size p, we direct it by choosing a vertex, say r, as a root. With such a directed tree we associate a map Φr. For a random vector S having a p-variate tree-Kummer distribution and any root r, we prove that Φr(S) has independent components. Moreover, we show that if S is a random vector in (0,∞)p and for any leaf r of the tree the components of Φr(S) are independent, then one of these components has a Gamma distribution and the remaining p−1 components have Kummer distributions. Our point of departure is a relatively simple independence property due to Hamza and Vallois (2016). It states that if X and Y are independent random variables having Kummer and Gamma distributions (with suitably related parameters) and T:(0,∞)2→(0,∞)2 is the involution defined by T(x,y)=(y/(1+x),x+xy/(1+x)), then the random vector T(X,Y) has also independent components with Kummer and gamma distributions. By a method inspired by a proof of a similar result for the MY property, we show that this independence property characterizes the gamma and Kummer laws.

Suggested Citation

  • Piliszek, Agnieszka & Wesołowski, Jacek, 2016. "Kummer and gamma laws through independences on trees—Another parallel with the Matsumoto–Yor property," Journal of Multivariate Analysis, Elsevier, vol. 152(C), pages 15-27.
    • Handle: RePEc:eee:jmvana:v:152:y:2016:i:c:p:15-27
    DOI: 10.1016/j.jmva.2016.07.004
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    References listed on IDEAS

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    1. Matsumoto, Hiroyuki & Wesolowski, Jacek & Witkowski, Piotr, 2009. "Tree structured independence for exponential Brownian functionals," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3798-3815, October.
    2. Ronald W. Butler, 1998. "Generalized Inverse Gaussian Distributions and their Wishart Connections," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 25(1), pages 69-75, March.
    3. Bobecka, Konstancja, 2015. "The Matsumoto–Yor property on trees for matrix variates of different dimensions," Journal of Multivariate Analysis, Elsevier, vol. 141(C), pages 22-34.
    4. Koudou, Angelo Efoevi, 2012. "A Matsumoto–Yor property for Kummer and Wishart random matrices," Statistics & Probability Letters, Elsevier, vol. 82(11), pages 1903-1907.
    5. Massam, Hélène & Wesolowski, Jacek, 2006. "The Matsumoto-Yor property and the structure of the Wishart distribution," Journal of Multivariate Analysis, Elsevier, vol. 97(1), pages 103-123, January.
    6. Wesolowski, Jacek & Witkowski, Piotr, 2007. "Hitting times of Brownian motion and the Matsumoto-Yor property on trees," Stochastic Processes and their Applications, Elsevier, vol. 117(9), pages 1303-1315, September.
    7. Hamza, Marwa & Vallois, Pierre, 2016. "On Kummer’s distribution of type two and a generalized beta distribution," Statistics & Probability Letters, Elsevier, vol. 118(C), pages 60-69.
    8. Matsumoto, Hiroyuki & Yor, Marc, 2003. "Interpretation via Brownian motion of some independence properties between GIG and gamma variables," Statistics & Probability Letters, Elsevier, vol. 61(3), pages 253-259, February.
    9. Stirzaker, David, 2005. "Stochastic Processes and Models," OUP Catalogue, Oxford University Press, number 9780198568148.
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