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.References
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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