The Lukacs–Olkin–Rubin theorem without invariance of the ``quotient''

Abstract

Abstract. The Lukacs theorem is one of the most brilliant results in the area of characterizations of probability distributions. First, because it gives a deep insight into the nature of independence properties of the gamma distribution; second, because it uses beautiful and non-trivial mathematics. Originally it was proved for probability distributions concentrated on (0, ∞). In 1962 Olkin and Rubin extended it to matrix variate distributions. Since that time it has been believed that the fundamental reason such an extension is possible, is the assumed property of invariance of the distribution of the "quotient" (properly defined for matrices). The main result of this paper is that the matrix variate Lukacs theorem holds without any invariance assumption for the "quotient". The argument is based on solutions of some functional equations in matrix variate real functions, which seem to be of independent interest. The proofs use techniques of differential calculus in the cone of positive definite symmetric matrices.

Multiplicative Cauchy functional equation on symmetric cones

“…Functional equations for w (1) were already considered in [2] for differentiable functions and in [16] for continuous functions of real or complex Hermitian positive definite matrices of rank strictly greater than 2. Without any regularity assumptions it was solved on the Lorentz cone [18].…”

“…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].…”

“…The first one is connected to the Jordan triple product: w (1) x = x 1/2 , where x 1/2 is the unique symmetric positive definite square root of x = x 1/2 · x 1/2 . The second is w = t x , where t x is the lower triangular matrix in the Cholesky decomposition of x.…”

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

“…Bobecka and Wesołowski [2] assuming existence of strictly positive, twice differentiable densities proved a characterization of Wishart distribution on the cone + for division algorithm g 1 (a) = a −1/2 , where a 1/2 denotes the unique positive definite symmetric root of a ∈ + . These results were generalized to all non-octonion symmetric cones of rank >2 and to the Lorentz cone for strictly positive and continuous densities by Kołodziejek [12,13].…”

“…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.…”