## Borel theorems for random matrices from the classical compact symmetric spaces.(English)Zbl 1149.15016

The paper deals with Borel theorems for random matrices. The authors study random vectors of the form
$(\text{Tr}(A^{(1)}V), \text{Tr}(A^{(2)}V), \ldots, \text{Tr}(A^{(r)}V)),$
where $$V$$ is a uniformly distributed element of a matrix version of a classical compact symmetric space, and $$A^{(i)}$$, $$i=1,2,\dots,r$$, are deterministic parameter matrices. They show that for increasing matrix sizes these random vectors converge to a joint Gaussian limit, and compute its covariances. This generalizes previous results of P. Diaconis and S. N. Evans [Trans. Am. Math. Soc. 353, No. 7, 2615–2633 (2001; Zbl 1008.15013)] for Haar distributed matrices from the classical compact groups.
The authors review the extended Wick calculus and explain its connection with the Diaconis-Shahshahani integral [P. Diaconis and M. Shahshahani, Applied Probability Trust, 49–62 (1954; Zbl 0807.15015)]. The proof of the main results uses these tools and the integration formulas, due to B. Collins and P. Ṡniady [Commun. Math. Phys. 264, No. 3, 773–795 (2006; Zbl 1108.60004)], for polynomial functions on the classical compact groups.

### MSC:

 15B52 Random matrices (algebraic aspects) 60F05 Central limit and other weak theorems 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 43A75 Harmonic analysis on specific compact groups

### Citations:

Zbl 1008.15013; Zbl 0807.15015; Zbl 1108.60004
Full Text:

### References:

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