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# On Random Matrices from the Compact Classical Groups

Kurt Johansson
Annals of Mathematics
Second Series, Vol. 145, No. 3 (May, 1997), pp. 519-545
DOI: 10.2307/2951843
Stable URL: http://www.jstor.org/stable/2951843
Page Count: 27
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## Abstract

If M is a matrix taken randomly with respect to normalized Haar measure on U(n), O(n) or Sp(n), then the real and imaginary parts of the random variables Tr(Mk), k ≥ 1, converge to independent normal random variables with mean zero and variance k/2, as the size n of the matrix goes to infinity. For the unitary group this is a direct consequence of the strong Szego theorem for Toeplitz determinants. We will prove a conjecture of Diaconis saying that for U(n) the rate of convergence to the limiting normal is O(n-δ n) for some $\delta > 0$, and for O(n) and Sp(n) it is O(e-cn) for some $c > 0$.

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