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

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

Citations:

Zbl 0627.43009
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References:

[1] Farrell, R. H., (Multivariate Calculation (1985), Springer-Verlag: Springer-Verlag New York) · Zbl 0561.62048
[2] Gross, K. I.; Richards, D. St. P., Special functions of matrix argument. I. Algebraic induction, zonal polynomials and hypergeometric functions, Trans. Amer. Math. Soc., 301, 781-811 (1987) · Zbl 0626.33010
[3] Helgason, S., (Groups and Geometric Analysis (1984), Academic Press: Academic Press New York)
[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)
[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
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