A central limit theorem on the space of positive definite symmetric matrices. (English) Zbl 0736.60025

A central limit theorem is proved on the space \({\mathcal P}_ n\) of positive definite symmetric matrices. To do this, some natural analogs of the mean and dispersion on \({\mathcal P}_ n\) are defined and investigated. One uses a Taylor expansion of the spherical functions on \({\mathcal P}_ n\).


60F05 Central limit and other weak theorems
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
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[1] [1] , Dispersion d’une mesure de probabilité sur SL(2,ℝ) biinvariante par SO(2,ℝ) et théorème de la limite centrale, exposé Oberwolfach, 1975.
[2] [2] , , Jordan Algebras, Symmetric Cones and Symmetric Domains, to appear.
[3] [3] , Isotropic infinitely divisible measures on symmetric spaces, Acta Math., 111 (1964), 213-246. · Zbl 0154.43804
[4] [4] , Groups and Geometric Analysis, Academic Press, New York, 1984.
[5] [5] , Semi-groups of measures on Lie groups, Trans. Amer. Math. Soc., 81 (1956), 264-293. · Zbl 0073.12402
[6] [6] , , , Limit theorems for the compositions of distributions in the Lobachevsky plane and space, Theory Prob. Appl., 4 (1959), 399-402.
[7] [7] , On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. Ecole Norm. Sup., 6 (1973), 413-455. · Zbl 0293.22019
[8] [8] , The Central Limit Theorem on Spaces of Positive Definite Matrices, J. Multivariate Anal., 29 (1989), 326-332. · Zbl 0681.60026
[9] [9] , Noneuclidean Harmonic Analysis, the Central Limit Theorem and Long Transmission Lines with Random Inhomogeneities, J. Multivariate Anal., 15 (1984), 261-276. · Zbl 0551.60022
[10] Bourbaki, N., Éléments de mathématique. Fasc. XXXI. Algèbre commutative. Chapitre 7: Diviseurs (1965) · Zbl 0128.01404
[11] [11] , Harmonic Analysis on Symmetric Spaces and Applications II, Springer-Verlag, New York, 1988. · Zbl 0668.10033
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