Electronic Journal of Probability
- Electron. J. Probab.
- Volume 16 (2011), paper no. 86, 2375-2405.
Stein's Method and the Multivariate CLT for Traces of Powers on the Compact Classical Groups
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.
Electron. J. Probab., Volume 16 (2011), paper no. 86, 2375-2405.
Accepted: 22 November 2011
First available in Project Euclid: 1 June 2016
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
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]
This work is licensed under aCreative Commons Attribution 3.0 License.
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