A central limit theorem on the space of positive definite symmetric matrices

Piotr Graczyk

Annales de l'institut Fourier (1992)

  • Volume: 42, Issue: 4, page 857-874
  • ISSN: 0373-0956

Abstract

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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 .

How to cite

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Graczyk, Piotr. "A central limit theorem on the space of positive definite symmetric matrices." Annales de l'institut Fourier 42.4 (1992): 857-874. <http://eudml.org/doc/74976>.

@article{Graczyk1992,
abstract = {A central limit theorem is proved on the space $\{\cal P\}_ n$ of positive definite symmetric matrices. To do this, some natural analogs of the mean and dispersion on $\{\cal P\}_ n$ are defined and investigated. One uses a Taylor expansion of the spherical functions on $\{\cal P\}_ n$.},
author = {Graczyk, Piotr},
journal = {Annales de l'institut Fourier},
keywords = {central limit theorem; spherical functions; symmetric spaces},
language = {eng},
number = {4},
pages = {857-874},
publisher = {Association des Annales de l'Institut Fourier},
title = {A central limit theorem on the space of positive definite symmetric matrices},
url = {http://eudml.org/doc/74976},
volume = {42},
year = {1992},
}

TY - JOUR
AU - Graczyk, Piotr
TI - A central limit theorem on the space of positive definite symmetric matrices
JO - Annales de l'institut Fourier
PY - 1992
PB - Association des Annales de l'Institut Fourier
VL - 42
IS - 4
SP - 857
EP - 874
AB - A central limit theorem is proved on the space ${\cal P}_ n$ of positive definite symmetric matrices. To do this, some natural analogs of the mean and dispersion on ${\cal P}_ n$ are defined and investigated. One uses a Taylor expansion of the spherical functions on ${\cal P}_ n$.
LA - eng
KW - central limit theorem; spherical functions; symmetric spaces
UR - http://eudml.org/doc/74976
ER -

References

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  1. [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. [2] J. FARAUT, A. KORANYI, Jordan Algebras, Symmetric Cones and Symmetric Domains, to appear. 
  3. [3] R. GANGOLLI, Isotropic infinitely divisible measures on symmetric spaces, Acta Math., 111 (1964), 213-246. Zbl0154.43804MR28 #4557
  4. [4] S. HELGASON, Groups and Geometric Analysis, Academic Press, New York, 1984. Zbl0543.58001
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  6. [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. Zbl0107.35603
  7. [7] B. KOSTANT, On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. Ecole Norm. Sup., 6 (1973), 413-455. Zbl0293.22019MR51 #806
  8. [8] D.St.P. RICHARDS, The Central Limit Theorem on Spaces of Positive Definite Matrices, J. Multivariate Anal., 29 (1989), 326-332. Zbl0681.60026MR91a:60031
  9. [9] A. TERRAS, Noneuclidean Harmonic Analysis, the Central Limit Theorem and Long Transmission Lines with Random Inhomogeneities, J. Multivariate Anal., 15 (1984), 261-276. Zbl0551.60022MR86k:43009
  10. [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. Zbl0627.43009MR88j:43006
  11. [11] A. TERRAS, Harmonic Analysis on Symmetric Spaces and Applications II, Springer-Verlag, New York, 1988. Zbl0668.10033MR89k:22017

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