The Lukacs–Olkin–Rubin theorem without invariance of the “quotient”

Konstancja Bobecka ¹   ; Jacek Weso/lowski ²

¹ Faculty of Mathematics and Information Science Warsaw University of Technology Pl. Politechniki 1 00-661 Warszawa, Poland
² Faculty of Mathematics and Information Science Warsaw University of Technologcr Pl. Politechniki 1 00-661 Warszawa, Poland

Studia Mathematica, Tome 152 (2002) no. 2, pp. 147-160

Résumé

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 $(0,\infty )$. 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.

Détail

DOI : 10.4064/sm152-2-4

Keywords: lukacs theorem brilliant results area characterizations probability distributions first because gives deep insight nature independence properties gamma distribution second because uses beautiful non trivial mathematics originally proved probability distributions concentrated infty olkin rubin extended matrix variate distributions since time has believed fundamental reason extension possible assumed property invariance distribution quotient properly defined matrices main result paper matrix variate lukacs theorem holds without invariance assumption quotient argument based solutions functional equations matrix variate real functions which seem independent interest proofs techniques differential calculus cone positive definite symmetric matrices