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Linear functions on the classical matrix groups

Author: Elizabeth Meckes
Journal: Trans. Amer. Math. Soc. 360 (2008), 5355-5366
MSC (2000): Primary 60F05; Secondary 60B15, 60B10
Published electronically: May 20, 2008
MathSciNet review: 2415077
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Abstract: Let $ M$ be a random matrix in the orthogonal group $ \mathcal{O}_n$, distributed according to Haar measure, and let $ A$ be a fixed $ n\times n$ matrix over $ \mathbb{R}$ such that $ \mathrm{Tr}(AA^t)=n$. Then the total variation distance of the random variable $ \mathrm{Tr}(AM)$ to a standard normal random variable is bounded by $ \frac{2\sqrt{3}} {n-1}$, and this rate is sharp up to the constant. Analogous results are obtained for $ M$ a random unitary matrix and $ A$ a fixed $ n\times n$ matrix over $ \mathbb{C}$. The proofs are applications of a new abstract normal approximation theorem which extends Stein's method of exchangeable pairs to situations in which continuous symmetries are present.

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Additional Information

Elizabeth Meckes
Affiliation: Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106
Address at time of publication: American Institute of Mathematics, 360 Portage Avenue, Palo Alto, California 94306

Keywords: Stein's method, normal approximation, random matrices
Received by editor(s): September 22, 2006
Published electronically: May 20, 2008
Additional Notes: This research was supported in part by the ARCS Foundation.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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