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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The Berry-Esseen bound for character ratios
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by Qi-Man Shao and Zhong-Gen Su PDF
Proc. Amer. Math. Soc. 134 (2006), 2153-2159 Request permission

Abstract:

Let $\lambda$ be a partition of $n$ chosen from the Plancherel measure of the symmetric group $S_n$, let $\chi ^\lambda (12)$ be the irreducible character of the symmetric group parameterized by $\lambda$ evaluated on the transposition $(12)$, and let $\dim (\lambda )$ be the dimension of the irreducible representation parameterized by $\lambda$. Fulman recently obtained the convergence rate of $O(n^{-s})$ for any $0< s<\frac 12$ in the central limit theorem for character ratios ${(n-1) \over \sqrt {2} } {\chi ^\lambda (12) \over \dim (\lambda )}$ by developing a connection between martingale and character ratios, and he conjectures that the correct speed is $O(n^{-1/2})$. In this paper we confirm the conjecture via a refinement of Stein’s method for exchangeable pairs.
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Additional Information
  • Qi-Man Shao
  • Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403 – and – Department of Mathematics, Department of Statistics and Applied Probability, National University of Singapore
  • Email: qmshao@darkwing.uoregon.edu
  • Zhong-Gen Su
  • Affiliation: Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310027, People’s Republic of China
  • Received by editor(s): September 28, 2004
  • Received by editor(s) in revised form: February 4, 2005
  • Published electronically: December 19, 2005
  • Additional Notes: The first author was supported in part by Grant R-1555-000-035-112 at the National University of Singapore
    The second author was supported in part by NFS of China (No. 10371109)
  • Communicated by: Richard C. Bradley
  • © Copyright 2005 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 2153-2159
  • MSC (2000): Primary 60F05; Secondary 05E10, 60C05
  • DOI: https://doi.org/10.1090/S0002-9939-05-08177-3
  • MathSciNet review: 2215787