Electronic Journal of Probability

Stein's Method and the Multivariate CLT for Traces of Powers on the Compact Classical Groups

Christian Döbler and Michael Stolz

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Let $M$ 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 of traces of consecutive powers of $M$ 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.

Article information

Electron. J. Probab., Volume 16 (2011), paper no. 86, 2375-2405.

Accepted: 22 November 2011
First available in Project Euclid: 1 June 2016

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Zentralblatt MATH identifier

Primary: 15B52: Random matrices
Secondary: 60F05: Central limit and other weak theorems 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60]

random matrices compact Lie groups Haar measure traces of powers Stein's method normal approximation exchangeable pairs heat kernel power sum symmetric polynomials

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Döbler, Christian; Stolz, Michael. Stein's Method and the Multivariate CLT for Traces of Powers on the Compact Classical Groups. Electron. J. Probab. 16 (2011), paper no. 86, 2375--2405. doi:10.1214/EJP.v16-960. https://projecteuclid.org/euclid.ejp/1464820255

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