Stein's method and the multivariate CLT for traces of powers on the classical compact groups

Details Stein's method and the multivariate CLT for traces of powers on the classical compact groups Stein's method and the multivariate CLT for traces of powers on the classical compact groups

2010-12-16

arxiv.org/abs/1012.3730v2

Mathematics

Probability

Main result strengthened, proof of Prop. 1.3 added

Scientific paper

Let $M_n$ be a random element of the unitary, special orthogonal, or unitary symplectic groups, distributed according to Haar measure. By a classical result of Diaconis and Shahshahani, for large matrix size $n$, the vector $ (\on{Tr}(M_n), \on{Tr}(M_n^2),..., \on{Tr}(M_n^d))$ tends to a vector of independent (real or complex) Gaussian random variables. Recently, Jason Fulman has demonstrated that for a single power $j$ (which may grow with $n$), a speed of convergence result may be obtained via Stein's method of exchangeable pairs. In this note, we extend Fulman's result to the multivariate central limit theorem for the full vector of traces of powers.

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