The Matsumoto–Yor Property and Its Converse on Symmetric Cones

Kołodziejek, Bartosz

doi:10.1007/s10959-015-0648-2

Cited by 8 publications

(9 citation statements)

References 18 publications

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“…Note that in the case of the tree T which is a 1-2, chain the above density agrees with (11). Thus, as observed in Section 3.1, the transformations Φ 1 and Φ 2 applied to a bivariate random vector with such density produce random vectors with independent Kummer and Gamma components.…”

Section: 3supporting

confidence: 70%

“…Assume that a random vector (S 1 , S 2 ) has the density p as in Eq. (11). Define (X 1,(1) , X 2,(1) ) = Φ 1 (S 1 , S 2 ) and (X 1,(2) , X 2,(2) ) = Φ 2 (S 1 , S 2 ).…”

Section: Independence Properties Of Tree-kummer Distributionmentioning

confidence: 99%

“…We will consider a more general distribution of this type by introducing three positive parameters c 1 , c 2 and c 1,2 and writing (11) p(s 1 , s 2 ) ∝ s a1−1…”

Section: Kummer and Gamma Characterizationmentioning

confidence: 99%

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

Piliszek¹,

Wesołowski²

2016

Journal of Multivariate Analysis

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Abstract. The paper develops a rather unexpected parallel to the multivariate Matsumoto-Yor (MY) property on trees considered in [19]. 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 [10]. 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.

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Section: 3supporting

confidence: 70%

“…Assume that a random vector (S 1 , S 2 ) has the density p as in Eq. (11). Define (X 1,(1) , X 2,(1) ) = Φ 1 (S 1 , S 2 ) and (X 1,(2) , X 2,(2) ) = Φ 2 (S 1 , S 2 ).…”

Section: Independence Properties Of Tree-kummer Distributionmentioning

confidence: 99%

See 1 more Smart Citation

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

Piliszek¹,

Wesołowski²

2016

Journal of Multivariate Analysis

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“…Let us mention that the MY property and related characterization triggered a lot of further research developing in several directions: (1) more general algebraic structures as, a multivariate tree-generated version in Massam and Wesołowski (2004), matrix variate versions in Letac and Wesołowski (2000), Wesołowski (2002), Massam and Wesołowski (2006), a combination of the matrix variate and multivariate tree-generated setting in Bobecka (2015), symmetric cone variate in Kołodziejek (2017), a version in free probability in Szpojankowski (2017);…”

Section: Gmy Property For Gig Matricesmentioning

confidence: 99%

About an extension of the Matsumoto-Yor property

Letac¹,

Wesołowski²

2022

Preprint

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If α, β > 0 are distinct and if A and B are independent non-degenerate positive random variables such that S = 1 B βA+B αA+B and T = 1 A βA+B αA+Bare independent, we prove that this happens if and only if the A and B have generalized inverse Gaussian distributions with suitable parameters. Essentially, this has already been proved in Bao and Noack ( 2021) with supplementary hypothesis on existence of smooth densities. The sources of these questions are an observation about independence properties of the exponential Brownian motion due to Matsumoto and Yor ( 2001) and a recent work of Croydon and Sasada (2000) on random recursion models rooted in the discrete Korteweg -de Vries equation, where the above result was conjectured.We also extend the direct result to random matrices proving that a matrix variate analogue of the above independence property is satisfied by independent matrix-variate GIG variables. The question of characterization of GIG random matrices through this independence property remains open.

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“…In the same paper authors generalized the Matsumoto-Yor property to the framework of real symmetric matrices. Further generalizations of different nature can be found for example in [14], [1] and [11].…”

Section: Introductionmentioning

confidence: 99%