## Linear functions on the classical matrix groups

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**360**(2008), 5355-5366 Request permission

## 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.## References

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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
- Received by editor(s): September 22, 2006
- Published electronically: May 20, 2008
- Additional Notes: This research was supported in part by the ARCS Foundation.
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2415077