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A Matsumoto–Yor property for Kummer and Wishart random matrices

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  • Koudou, Angelo Efoevi

Abstract

For a positive integer r, let I denote the r×r unit matrix. Let X and Y be two independent r×r real symmetric and positive definite random matrices. Assume that X follows a Kummer distribution while Y follows a non-degenerate Wishart distribution, with suitable parameters. This note points out the following observation: the random matrices U:=[I+(X+Y)−1]1/2[I+X−1]−1[I+(X+Y)−1]1/2 and V:=X+Y are independent and U follows a matrix beta distribution while V follows a Kummer distribution. This generalizes to the matrix case an independence property established in Koudou and Vallois (2010) for r=1.

Suggested Citation

  • Koudou, Angelo Efoevi, 2012. "A Matsumoto–Yor property for Kummer and Wishart random matrices," Statistics & Probability Letters, Elsevier, vol. 82(11), pages 1903-1907.
  • Handle: RePEc:eee:stapro:v:82:y:2012:i:11:p:1903-1907
    DOI: 10.1016/j.spl.2012.06.024
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    References listed on IDEAS

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    1. 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.
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    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. Wesołowski, Jacek, 2015. "On the Matsumoto–Yor type regression characterization of the gamma and Kummer distributions," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 145-149.

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