The Lukacs-Olkin-Rubin theorem without invariance of the "quotient"

Konstancja Bobecka, and Jacek Wesołowski. "The Lukacs-Olkin-Rubin theorem without invariance of the "quotient"." Studia Mathematica 152.2 (2002): 147-160. <http://eudml.org/doc/285119>.

@article{KonstancjaBobecka2002,
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.},
author = {Konstancja Bobecka, Jacek Wesołowski},
journal = {Studia Mathematica},
keywords = {Wishart distribution; positive definite symmetric matrices; independence; random matrices; matrix variate functions; functional equations},
language = {eng},
number = {2},
pages = {147-160},
title = {The Lukacs-Olkin-Rubin theorem without invariance of the "quotient"},
url = {http://eudml.org/doc/285119},
volume = {152},
year = {2002},
}

TY - JOUR
AU - Konstancja Bobecka
AU - Jacek Wesołowski
TI - The Lukacs-Olkin-Rubin theorem without invariance of the "quotient"
JO - Studia Mathematica
PY - 2002
VL - 152
IS - 2
SP - 147
EP - 160
AB - 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.
LA - eng
KW - Wishart distribution; positive definite symmetric matrices; independence; random matrices; matrix variate functions; functional equations
UR - http://eudml.org/doc/285119
ER -