Matrix Measures, Random Moments, and Gaussian Ensembles |
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Authors: | Holger Dette Jan Nagel |
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Institution: | (1) School of Mathematics, University of Southampton, Highfield, SO17 1BJ, UK;(2) Fakult?t f?r Mathematik, Ruhr-Universit?t Bochum, 44780 Bochum, Germany; |
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Abstract: | We consider the moment space Mn\mathcal{M}_{n} corresponding to p×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 (S1,n, ... , Sn,n)* ~ U (Mn)(S_{1,n}, \dots , S_{n,n})^{*} \sim\mathcal{U} (\mathcal{M}_{n}) are studied as n→∞. In particular, it is shown that an appropriately centered and standardized version of the vector (S
1,n
,…,S
k,n
)∗ converges weakly to a vector of k independent p×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 Mn\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. |
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Keywords: | |
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