A Characterization of the Riesz Probability Distribution

The Lukacs–Olkin–Rubin Theorem on Symmetric Cones Without Invariance of the “Quotient”

“…We consider quotient U for an arbitrary, fixed division algorithm g as in the original paper of Olkin and Rubin [21], additionally satisfying some natural conditions. In the known cases (g = g 1 and g = g 2 ), this improves the results obtained in Bobecka and Wesołowski, Hassairi et al and Kołodziejek [2,11,13]. In general case, the densities of X and Y are given in terms of, so-called, w-multiplicative Cauchy functions, that is functions satisfying…”

“…Exploiting the same approach, with the same technical assumptions on densities as in Bobecka and Wesołowski [2], it was proven by Hassairi et al [11] that the independence of X + Y and the quotient defined through the Cholesky decomposition, i.e., g 2 (a) = T −1 a , where T a is a lower triangular matrix such that a = T a · T T a ∈ + , characterizes a wider family of distributions called Riesz (or sometimes called RieszWishart). This fact shows that the invariance property assumed in Olkin and Rubin [21] and Casalis and Letac [7] is not of technical nature only.…”

“…Section 3 is devoted to consideration of w-logarithmic Cauchy functions and the Olkin-Baker functional equation. In that section, we offer much shorter, simpler and covering more general cones proof of the Olkin-Baker functional equation than given in Bobecka and Wesołowski, Hassairi et al and Kołodziejek [2,11,13].…”

Multiplicative Cauchy functional equation on symmetric cones

“…Our interest in this functional equation stems from investigations of characterization problems for probabilistic measures concentrated on Ω + or more generally on symmetric cones-see [1,3,4,10,12,13].…”

“…It was indirectly solved for differentiable functions in [10]. Here we solve this functional equation under no regularity assumptions and, at the same time, we find all real characters of the triangular group.…”