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The Berry-Esseen Bound for Character Ratios

Qi-Man Shao and Zhong-Gen Su
Proceedings of the American Mathematical Society
Vol. 134, No. 7 (Jul., 2006), pp. 2153-2159
Stable URL: http://www.jstor.org/stable/4098248
Page Count: 7
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
The Berry-Esseen Bound for Character Ratios
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Abstract

Let λ be a partition of n chosen from the Plancherel measure of the symmetric group Sn, let $\chi^{\lambda}(12)$ be the irreducible character of the symmetric group parameterized by λ evaluated on the transposition (12), and let $dim(\lambda)$ be the dimension of the irreducible representation parameterized by λ. Fulman recently obtained the convergence rate O(n-s) for any $0 < s < \frac{1}{2}$ in the central limit theorem for character ratios $\frac{(n-1)}{\sqrt{2}} \frac{\xi^{\lambda} (12)}{\dim (\lambda)}$ 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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