The central limit theorem on spaces of positive definite matrices (Q1823538)

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 \textit{A. Terras} [ibid. 23, 13-36 (1987; Zbl 0627.43009)].