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Let M be a random matrix chosen from Haar measure on the unitary group U n . Let Z = X + iY be a standard complex normal random variable with X and Y independent, mean 0 and variance ½ normal variables. We show that for j = 1, 2, …, Tr( M j ) are independent and distributed as √ jZ asymptotically as n →∞. This result is used to study the set of eigenvalues of M. Similar results are given for the orthogonal and symplectic and symmetric groups.
Diaconis et al. (Sat,) studied this question.
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