A central limit theorem on the space of positive definite symmetric matrices
Graczyk, Piotr
Annales de l'Institut Fourier, Tome 42 (1992), p. 857-874 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

On démontre un théorème de la limite centrale sur l’espace des matrices symétriques définies positives. Dans ce but on introduit et étudie certains analogues de la moyenne et de la dispersion d’une mesure. On utilise un développent de Taylor des fonctions sphériques sur l’espace considéré.

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

@article{AIF_1992__42_4_857_0,
     author = {Graczyk, Piotr},
     title = {A central limit theorem on the space of positive definite symmetric matrices},
     journal = {Annales de l'Institut Fourier},
     volume = {42},
     year = {1992},
     pages = {857-874},
     doi = {10.5802/aif.1312},
     mrnumber = {93m:60023},
     zbl = {0736.60025},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1992__42_4_857_0}
}

Graczyk, Piotr. A central limit theorem on the space of positive definite symmetric matrices. Annales de l'Institut Fourier, Tome 42 (1992) pp. 857-874. doi : 10.5802/aif.1312. http://gdmltest.u-ga.fr/item/AIF_1992__42_4_857_0/

[1] J. Faraut, 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] J. Faraut, A. Koranyi, Jordan Algebras, Symmetric Cones and Symmetric Domains, to appear.

[3] R. Gangolli, Isotropic infinitely divisible measures on symmetric spaces, Acta Math., 111 (1964), 213-246. | MR 28 #4557 | Zbl 0154.43804

[4] S. Helgason, Groups and Geometric Analysis, Academic Press, New York, 1984. | Zbl 0543.58001

[5] G.A. Hunt, Semi-groups of measures on Lie groups, Trans. Amer. Math. Soc., 81 (1956), 264-293. | MR 18,54a | Zbl 0073.12402

[6] F.I. Karpelevich, V.N. Tutubalin, M.G. Shur, Limit theorems for the compositions of distributions in the Lobachevsky plane and space, Theory Prob. Appl., 4 (1959), 399-402.

[7] B. Kostant, On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. Ecole Norm. Sup., 6 (1973), 413-455. | Numdam | MR 51 #806 | Zbl 0293.22019

[8] D.St.P. Richards, The Central Limit Theorem on Spaces of Positive Definite Matrices, J. Multivariate Anal., 29 (1989), 326-332. | MR 91a:60031 | Zbl 0681.60026

[9] A. Terras, Noneuclidean Harmonic Analysis, the Central Limit Theorem and Long Transmission Lines with Random Inhomogeneities, J. Multivariate Anal., 15 (1984), 261-276. | MR 86k:43009 | Zbl 0551.60022

[10] A. Terras, Asymptotics of Special Functions and the Central Limit Theorem on the Space Pn of Positive n × n Matrices, J. Multivariate Anal., 23 (1987), 13-36. | MR 88j:43006 | Zbl 0627.43009

[11] A. Terras, Harmonic Analysis on Symmetric Spaces and Applications II, Springer-Verlag, New York, 1988. | MR 89k:22017 | Zbl 0668.10033