## Electronic Journal of Probability

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

#### Abstract

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

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

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

https://projecteuclid.org/euclid.ejp/1464820255

Digital Object Identifier
doi:10.1214/EJP.v16-960

Mathematical Reviews number (MathSciNet)
MR2861678

Zentralblatt MATH identifier
1243.15023

Rights

#### Citation

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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