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

Abstract

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.

Main result strengthened, proof of Prop. 1.3 added

References in corpus (2)

• Multivariate normal approximation with Stein's method of exchangeable pairs under a general linearity condition
• Multivariate normal approximation using exchangeable pairs

Cited by in corpus (1)

• Fourth moment theorems on the Poisson space in any dimension