Matsumoto Yor Research Articles

About an extension of the Matsumoto–Yor property

About an extension of the Matsumoto–Yor property

Matsumoto–Yor and Dufresne type theorems for a random walk on positive definite matrices

Matsumoto–Yor and Dufresne type theorems for a random walk on positive definite matrices

A Matsumoto–Yor characterization for Kummer and Wishart random matrices

In the paper we resolve positively the conjecture on a characterization of matrix Kummer and Wishart laws through independence property, which was posed in Koudou (2012) [12]. Apart from the probabilistic result, we determine the general solution of the functional equation associated to the characterization problem under weak assumptions.

On the Matsumoto–Yor property in free probability

We study the Matsumoto–Yor property in free probability. We prove that the limiting empirical eigenvalue distribution of the GIG matrices and the Marchenko–Pastur distribution have the free Matsumoto–Yor property. Finally we characterize these distributions by regression properties in free probability.

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

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 Φ r . For a random vector S having a p -variate tree-Kummer distribution and any root r , we prove that Φ r ( 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 Φ r ( 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 + x y / ( 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.

The Matsumoto\u2013Yor Property and Its Converse on Symmetric Cones

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.

On the Matsumoto–Yor type regression characterization of the gamma and Kummer distributions

In this paper we study a Matsumoto–Yor type property for the gamma and Kummer independent variables discovered by Koudou and Vallois (2012). We prove that constancy of regressions of U=(1+(X+Y)−1)/(1+X−1) given V=X+Y and of U−1 given V, where X and Y are independent and positive random variables, characterizes the gamma and Kummer distributions. This result completes characterizations by independence of U and V obtained, under smoothness assumptions for densities, in Koudou and Vallois (2011, 2012). Since we work with differential equations for the Laplace transforms, no density assumptions are needed.

The Matsumoto–Yor property on trees for matrix variates of different dimensions

The paper is devoted to an extension of the multivariate Matsumoto–Yor (MY) independence property with respect to a tree with p vertices to the case where random variables corresponding to the vertices of the tree are replaced by random matrices. The converse of the p -variate MY property, which characterizes the product of one gamma and p − 1 generalized inverse Gaussian distributions, is extended to characterize the product of the Wishart and p − 1 matrix generalized inverse Gaussian distributions.

Independence properties of the Matsumoto–Yor type

We define Letac-Wesolowski-Matsumoto-Yor (LWMY) functions as decreasing functions from $(0,\infty)$ onto $(0,\infty)$ with the following property: there exist independent, positive random variables $X$ and $Y$ such that the variables $f(X+Y)$ and $f(X)-f(X+Y)$ are independent. We prove that, under additional assumptions, there are essentially four such functions. The first one is $f(x)=1/x$. In this case, referred to in the literature as the Matsumoto-Yor property, the law of $X$ is generalized inverse Gaussian while $Y$ is gamma distributed. In the three other cases, the associated densities are provided. As a consequence, we obtain a new relation of convolution involving gamma distributions and Kummer distributions of type 2.

Hitting times of Brownian motion and the Matsumoto–Yor property on trees

The Matsumoto–Yor property in the bivariate case was originally defined through properties of functionals of the geometric Brownian motion. A multivariate version of this property was described in the language of directed trees and outside of the framework of stochastic processes in Massam and Wesołowski [H. Massam, J. Wesołowski, The Matsumoto–Yor property on trees, Bernoulli 10 (2004) 685–700]. Here we propose its interpretation through properties of hitting times of Brownian motion, extending the interpretation given in the bivariate case in Matsumoto and Yor [H. Matsumoto, M. Yor, Interpretation via Brownian motion of some independence properties between GIG and gamma variables, Statist. Probab. Lett. 61 (2003) 253–259].

Multivariate Matsumoto–Yor property is rather restrictive

Matsumoto and Yor [2001. An analogue of Pitman's 2 M - X theorem for exponential Wiener functionals. Part II: the role of the GIG laws. Nagoya Math. J. 162, 65–86] discovered an interesting invariance property of a product of the generalized inverse Gaussian (GIG) and the gamma distributions. For univariate random variables or symmetric positive definite random matrices it is a characteristic property for this pair of distributions. It appears that for random vectors the Matsumoto–Yor property characterizes only very special families of multivariate GIG and gamma distributions: components of the respective random vectors are grouped into independent subvectors, each subvector having linearly dependent components. This complements the version of the multivariate Matsumoto–Yor property on trees and related characterization obtained in Massam and Wesołowski [2004. The Matsumoto–Yor property on trees. Bernoulli 10, 685–700].

A link between the Matsumoto–Yor property and an independence property on trees

We prove that an independence property established by Matsumoto and Yor [2001. An analogue of Pitman's 2 M - X theorem for exponential Wiener functional, Part II: the role of the generalized inverse Gaussian laws. Nagoya Math. J. 162, 65–86] and by Letac and Wesolowski [2000. An independence property for the product of GIG and gamma laws. Ann. Probab. 28, 1371–1383] is, in a particular case, a corollary of a result by Barndorff-Nielsen and Koudou [1998. Trees with random conductivities and the (reciprocal) inverse Gaussian distribution. Adv. Appl. Probab. 30, 409–424] where, for finite trees equipped with inverse Gaussian resistances, an exact distributional and independence result was established.

The Matsumoto–Yor property and the structure of the Wishart distribution

This paper establishes a link between a generalized matrix Matsumoto–Yor (MY) property and the Wishart distribution. This link highlights certain conditional independence properties within blocks of the Wishart and leads to a new characterization of the Wishart distribution similar to the one recently obtained by Geiger and Heckerman but involving independences for only three pairs of block partitionings of the random matrix. In the process, we obtain two other main results. The first one is an extension of the MY independence property to random matrices of different dimensions. The second result is its converse. It extends previous characterizations of the matrix generalized inverse Gaussian and Wishart seen as a couple of distributions. We present two proofs for the generalized MY property. The first proof relies on a new version of Herz's identity for Bessel functions of matrix arguments. The second proof uses a representation of the MY property through the structure of the Wishart.

The Matsumoto–Yor independence property for GIG and Gamma laws, revisited

Matsumoto and Yor have recently discovered an interesting invariance property of a product of the generalized inverse Gaussian and gamma distributions. In this paper we obtain: (1) a complete regression version of its converse; (2) a converse to the matrix variate Matsumoto–Yor property which extends an earlier result. Of independent interest is a functional equation for matrix valued functions, which has been solved in the course of investigation of the second problem.