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 | References | Similar Articles | Additional Information

Abstract: Let be a partition of chosen from the Plancherel measure of the symmetric group , let be the irreducible character of the symmetric group parameterized by evaluated on the transposition , and let be the dimension of the irreducible representation parameterized by . Fulman recently obtained the convergence rate of for any in the central limit theorem for character ratios by developing a connection between martingale and character ratios, and he conjectures that the correct speed is . 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

DOI:
http://dx.doi.org/10.1090/S0002-9939-05-08177-3

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