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

Abstract

We prove the Lukacs-Olkin-Rubin theorem without invariance of the distribution of the "quotient," which was the key assumption in the original proof of (Olkin-Rubin in Ann Math Stat 33: [1272][1273][1274][1275][1276][1277][1278][1279][1280] 1962). Instead, we assume existence of strictly positive continuous densities of respective random variables. We consider the (cone variate) "quotient" for any division algorithm satisfying some natural conditions. For that purpose, a new proof of the Olkin-Baker functional equation on symmetric cones is given.

The Matsumoto–Yor Property and Its Converse on Symmetric Cones

“…A new solution of a related functional equation on symmetric cones (see Theorem 4.5) was found under the assumption of continuity of respective functions with the use of the corresponding univariate result due to Weso lowski [18]. Similar reduction in regularity assumptions was recently performed in the density version of Lukacs-Olkin-Rubin in [7].…”

The generalized fundamental equation of information on symmetric cones

“…Inserting a multiplication algorithm g = w(y), y ∈ Ω, and x = e we obtain Det (w(y)) = (det y) dim Ω/r and hence det(w(y)x) = det y det x (8) for any x, y ∈ Ω.…”

“…An analogous problem, when the general form of the multiplication algorithm was considered for the Olkin-Baker functional equation was dealt with in [8], but that functional equation was much easier to solve. What is interesting, for scalar arguments, the fundamental equation of information can be brought to the Olkin-Baker equation (see the proof of the main theorem in [13]).…”

Characterization of beta distribution on symmetric cones

“…In the famous Lukacs-Olkin-Rubin Theorem (see [14] for Ω + case, [3] for all irreducible symmetric cones and [7,Remark 4.4] for its density version), the following independence property was analyzed: assume X and Y are independent random variables valued in Ω and V = X +Y and U = g(X +Y )X (here g = w −1 ) are also independent (supplemented with some technical assumptions). If the distribution of U is invariant under the group K of automorphisms, that is kU d = U for any k ∈ K, then X and Y follow Wishart distribution with the same scale parameter, regardless of the choice of multiplication algorithm w = g −1 .…”

“…Analogous characterization of Wishart distribution, when densities of respective random variables are given in terms of w-logarithmic functions is given in [7]. Unfortunately, we cannot answer the question whether there exists multiplication algorithm resulting in characterizing other distribution than beta or beta-Riesz.…”