Which distributions have the Matsumoto-Yor property?

Abstract

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.

A Matsumoto–Yor characterization for Kummer and Wishart random matrices

“…The independence property established in [12] has been proved for the case r = 1 in [13], where the converse has been proved too, thus providing a characterization of Kummer and gamma laws under the assumption of existence of smooth densities of X and Y . At the end of [12], the author writes that "It is highly likely, although not easy to prove, that this characterization holds also in the case of matrices."…”

“…From the probabilistic point of view, this family is somehow related to the so-called Matsumoto-Yor property. Koudou and Vallois in [14] (see also [13]) considered ψ (f ) (x, y) = (f (x + y), f (x) − f (x + y)) where f is some regular function and asked the following question: for which f there exist independent X and Y such that U and V are independent. The classical Matsumoto-Yor property (see [24,25]) is obtained for f (1)…”

On Kummer’s distribution of type two and a generalized beta distribution

“…Below, L(X) stands for the law of the random variable X. Let us recall a weak version of Theorem 2.1 in Koudou and Vallois (2011). Theorem 1.1 Let X and Y be two independent and positive random variables such that:…”

“…(1.4) Theorem 2.1 in Koudou and Vallois (2011) says more generally that if X and Y are independent r.v. 's whose log densities are locally integrable, then U and V defined by (1.3) are independent if and only if (1.2) holds.…”

“…Armero and Bayarri (1994) have considered the Gauss hypergeometric distribution GH(α, β, γ, z) := β 1+z (α, β, −γ) as a marginal prior distribution. However, in this paper, we conserve the notations of Koudou and Vallois (2011). More recently, Ristić, Popović and Nadarajah (2015) have introduced "skewed" distributions associated with β δ (a, b, c) and baseline cumulative distribution function F as cdf of the type :…”