A Matsumoto–Yor property for Kummer and Wishart random matrices

Koudou, Angelo Efoévi

doi:10.1016/j.spl.2012.06.024

Cited by 14 publications

(15 citation statements)

References 7 publications

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“…It is worth mentioning several related one-dimensional results [3,10] as well as results for random matrices [12,9].…”

Section: Introductionmentioning

confidence: 99%

The Matsumoto–Yor Property and Its Converse on Symmetric Cones

Kołodziejek

2015

J Theor Probab

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The Matsumoto-Yor (MY) property of the generalized inverse Gaussian and gamma distributions has many generalizations. As was observed in Letac and Wesołowski (Ann Probab 28:1371-1383, 2000, the natural framework for the multivariate MY property is symmetric cones; however, they prove their results for the cone of symmetric positive definite real matrices only. In this paper, we prove the converse to the symmetric cone-variate MY property, which extends some earlier results. The smoothness assumption for the densities of respective variables is reduced to continuity only. This enhancement was possible due to the new solution of a related functional equation for real functions defined on symmetric cones.

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“…It is worth mentioning several related one-dimensional results [3,10] as well as results for random matrices [12,9].…”

Section: Introductionmentioning

confidence: 99%

The Matsumoto–Yor Property and Its Converse on Symmetric Cones

Kołodziejek

2015

J Theor Probab

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“…First nontrivial observation to be made is that (9) solves (8). This was proved in [12] and follows directly by Proposition 2.1.…”

Section: Functional Equationsmentioning

confidence: 58%

“…In the present paper we will consider a generalization of f (3) to S + , which was introduced in [12]:…”

Section: Introductionmentioning

confidence: 99%

A Matsumoto–Yor characterization for Kummer and Wishart random matrices

Kołodziejek

2018

Journal of Mathematical Analysis and Applications

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“…The independence property (1) was extended to matrix variate Kummer and Wishart distributions in [14]. For more information on the matrix variate Kummer distribution one can consult [26] and Ch.…”

Section: Introductionmentioning

confidence: 99%

Change of measure technique in characterizations of the gamma and Kummer distributions

Piliszek

Wesołowski

2018

Journal of Mathematical Analysis and Applications

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Abstract. If X and Y are independent random variables with distributions µ nd ν then U = ψ(X, Y ) and V = φ(X, Y ) are also independent for some transformations ψ and φ. Properties of this type are known for many important probability distributions µ and ν. Also related characterization questions have been widely investigated. These are questions of the form: Let X and Y be independent and let U and V be also independent. Are the distributions of X and Y necessarily µ and ν, respectively?Recently two new properties and characterizations of this kind involving the Kummer distribution appeared in the literature. For independent X and Y with gamma and Kummer distributions Koudou and Vallois in [17] observed that U = (1 + (X + Y ) −1 )/(1 + X −1 ) and V = X + Y are also independent, and Hamza and Vallois in [13] observed that U = Y /(1 + X) and V = X(1 + Y /(1 + X)) are independent. In [16] and [17] characterizations related to the first property were proved, while the characterizations in the second setting have been recently given in [29]. These results were not fully satisfactory since in both cases technical assumptions on smoothness properties of densities of X and Y were needed. In [31], the assumption of independence of U and V in the first setting was weakened to constancy of regressions of U and U −1 given V with no density assumptions. However, the additional assumption E X −1 < ∞ was introduced.In the present paper we provide a complete answer to the characterization question in both settings without any additional technical assumptions regarding smoothness or existence of moments. The approach is, first, via characterizations exploiting some conditions imposed on regressions of U given V , which are weaker than independence, but for which moment assumptions are necessary. Second, using a technique of change of measure we show that the moment assumptions can be avoided.

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A Matsumoto–Yor property for Kummer and Wishart random matrices

Cited by 14 publications

References 7 publications

The Matsumoto–Yor Property and Its Converse on Symmetric Cones

The Matsumoto–Yor Property and Its Converse on Symmetric Cones

A Matsumoto–Yor characterization for Kummer and Wishart random matrices

Change of measure technique in characterizations of the gamma and Kummer distributions

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