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Matrix measures, random moments, and Gaussian ensembles. (English) Zbl 1237.60018

Summary: We consider the moment space \({\mathcal{M}}_{n}\) corresponding to \(p \times p\) real or complex matrix measures defined on the interval \([0,1]\). The asymptotic properties of the first \(k\) components of a uniformly distributed vector \((S_{1,n},\dots,S_{n,n})^{*} \sim \mathcal{U}({\mathcal{M}}_{n})\) are studied as \(n \to \infty\). In particular, it is shown that an appropriately centered and standardized version of the vector \((S_{1,n},\dots,S_{k,n })^{\ast}\) converges weakly to a vector of \(k\) independent \(p \times p\) Gaussian ensembles. For the proof of our results, we use some new relations between ordinary moments and canonical moments of matrix measures which are of their own interest. In particular, it is shown that the first \(k\) canonical moments corresponding to the uniform distribution on the real or complex moment space \({\mathcal{M}}_{n}\) are independent multivariate Beta-distributed random variables and that each of these random variables converges in distribution (as the parameters converge to infinity) to the Gaussian orthogonal ensemble or to the Gaussian unitary ensemble, respectively.

MSC:

60F05 Central limit and other weak theorems
15B52 Random matrices (algebraic aspects)
30E05 Moment problems and interpolation problems in the complex plane
60B20 Random matrices (probabilistic aspects)
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