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

DOI:
https://doi.org/10.1090/S0002-9947-08-04444-9

Published electronically:
May 20, 2008

MathSciNet review:
2415077

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Abstract | References | Similar Articles | Additional Information

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

MR Author ID:
754850

Email:
ese3@cwru.edu

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.