Abstract
In this article we obtain a superexponential rate of convergence in total variation between the traces of the first m powers of a random unitary matrices and a -dimensional Gaussian random variable. This generalizes previous results in the scalar case to the multivariate setting, and we also give the precise dependence on the dimensions m and n in the estimates with explicit constants. We are especially interested in the regime where m grows with n and our main result basically states that if , then the rate of convergence in the Gaussian approximation is times a correction. We also show that the Gaussian approximation remains valid for all without a fast rate of convergence.
Funding Statement
K.J. was supported by the grant KAW 2015.0270 from the Knut and Alice Wallenberg Foundation and the Swedish Research Council grant 2015-0487. G.L.’s research is supported by the SNSF Ambizione grant S-71114-05-01.
Acknowledgments
The authors would like to thank the anonymous referee for her/his constructive comments.
Citation
Kurt Johansson. Gaultier Lambert. "Multivariate normal approximation for traces of random unitary matrices." Ann. Probab. 49 (6) 2961 - 3010, November 2021. https://doi.org/10.1214/21-AOP1520
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