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The Berry-Esseen bound for character ratios

Authors: Qi-Man Shao and Zhong-Gen Su
Journal: Proc. Amer. Math. Soc. 134 (2006), 2153-2159
MSC (2000): Primary 60F05; Secondary 05E10, 60C05
Published electronically: December 19, 2005
MathSciNet review: 2215787
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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

Zhong-Gen Su
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310027, People’s Republic of China

Keywords: Berry-Esseen bound, character ratio, Plancherel measure, Stein's method.
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
Article copyright: © Copyright 2005 American Mathematical Society

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