We define Letac-Wesolowski-Matsumoto-Yor (LWMY) functions as decreasing functions from (0, ∞) onto (0, ∞) 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.

Lancaster probabilities, marginal distributions, orthogonal polynomials, extreme points of convex sets, moment sequences, 60-xx,

For four types of functions ξ : ]0, ∞[→]0, ∞[, we characterize the law of two independent and positive r.v.'s X and Y such that U := ξ(X + Y ) and V := ξ(X ) − ξ(X + Y ) are independent. The case ξ(x) = 1/x has been treated by Letac and Wesołowski (2000). As for the three other cases, under the weak assumption that X and Y have density functions whose logarithm is locally integrable, we prove that the distribution of (X , Y ) is unique. This leads to Kummer, gamma and beta distributions. This improves the result obtained in [1] where more regularity was required from the densities.

No 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.