The central limit theorem on spaces of positive definite matrices. (English) Zbl 0681.60026

Richards, Donald St. P.

J. Multivariate Anal. 29, No. 2, 326-332 (1989).

The linear group GL(n,\({\mathbb{R}})\) acts transitively on the space \({\mathcal P}_ n\) of positive \(n\times n\) matrices with isotopy group O(n). Hence \(G/O(n)\cong P_ n\). Let \(f\mapsto \hat f\) be the Helgason-Fourier transform for O(n)-invariant functions on \({\mathcal P}_ n\). O(n)-invariant functions, random variables respectively measures may be considered as the corresponding objects on the double-coset-space Gl(n,\({\mathbb{R}})//O(n).\)
The main result proved in this paper is a central-limit-type-theorem for O(n)-invariant r.v., which have a common density with respect to the invariant measure on \({\mathcal P}_ n:\) The suitably normalized random walk corresponding to an i.i.d. sequence converges weakly to a “Gaussian” distribution which is explicitly given via the Helgason-Fourier- transform. This result extends a CLT theorem of A. Terras [ibid. 23, 13-36 (1987; Zbl 0627.43009)].

Reviewer: W.Hazod

MSC:

60F05	Central limit and other weak theorems
43A65	Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
62E15	Exact distribution theory in statistics

Keywords:

positive definite matrices; central-limit-type-theorem; Helgason-Fourier- transform

Citations:

Zbl 0627.43009

References:

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[3]	Helgason, S., (Groups and Geometric Analysis (1984), Academic Press: Academic Press New York) · Zbl 0543.58001
[4]	Kushner, H. B., The linearization of the product of two zonal polynomials, SIAM J. Math. Anal., 19, 687-717 (1988) · Zbl 0642.33024
[5]	Terras, A., (Harmonic Analysis on Symmetric Spaces and Applications, I (1985), Springer-Verlag: Springer-Verlag New York) · Zbl 0574.10029
[6]	Terras, A., Asymptotics of special functions and the central limit theorem on the space \(P_n\) of positive \(n\) × \(n\) matrices, J. Multivariate Anal., 23, 13-36 (1987) · Zbl 0627.43009