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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Linear functions on the classical matrix groups
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by Elizabeth Meckes PDF
Trans. Amer. Math. Soc. 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