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The Lukacs–Olkin–Rubin theorem without invariance of the “quotient”. (English) Zbl 0993.62043
Summary: The Lukacs theorem [E. Lukacs, Ann. Math. Stat. 26, 319-324 (1955; Zbl 0065.11103)] 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 \((0,\infty)\). I. Olkin and H. Rubin [ibid. 33, 1272-1280 (1962; Zbl 0111.34202] 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.

MSC:
62H05 Characterization and structure theory for multivariate probability distributions; copulas
26B12 Calculus of vector functions
62E10 Characterization and structure theory of statistical distributions
26B99 Functions of several variables
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