Nonuniform Berry–Esseen bound for character ratios

Natthapol Dejtrakulwongse; Suporn Jongpreechaharn; Kritsana Neammanee; Natthapol Dejtrakulwongse; Suporn Jongpreechaharn; Kritsana Neammanee

doi:10.3934/math.2026933

AIMS Mathematics

2026, Volume 11, Issue 7: 23149-23172. doi: 10.3934/math.2026933

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Research article

Nonuniform Berry–Esseen bound for character ratios

1.
Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Pathumwan 10330, Bangkok, Thailand
2.
Center of Excellence in Mathematics, Ministry of Higher Education, Science, Research and Innovation, Ratchathewi 10400, Bangkok, Thailand

Received: 02 June 2026 Revised: 17 July 2026 Accepted: 24 July 2026 Published: 30 July 2026
MSC : 60F05, 20C30, 60J10

The first uniform bound in the normal approximation for character ratios was obtained by Fulman (2005) with the rate of convergence $ O(n^{-1/4}) $, and later improved to $ O(n^{-1/2}) $ by Shao and Su (2006). In this work, we use Stein's method and the Jack measure to derive a nonuniform Berry–Esseen-type bound for character ratios. Our result yields a sharper constant compared to previously known estimates. We also discuss an application of this approximation, particularly its role in random walk on the symmetric group $ S_n $.
- character ratio,
- normal approximation,
- uniform and nonuniform bounds,
- Stein's method and Jack measure
Citation: Natthapol Dejtrakulwongse, Suporn Jongpreechaharn, Kritsana Neammanee. Nonuniform Berry–Esseen bound for character ratios[J]. AIMS Mathematics, 2026, 11(7): 23149-23172. doi: 10.3934/math.2026933

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Abstract

The first uniform bound in the normal approximation for character ratios was obtained by Fulman (2005) with the rate of convergence $ O(n^{-1/4}) $, and later improved to $ O(n^{-1/2}) $ by Shao and Su (2006). In this work, we use Stein's method and the Jack measure to derive a nonuniform Berry–Esseen-type bound for character ratios. Our result yields a sharper constant compared to previously known estimates. We also discuss an application of this approximation, particularly its role in random walk on the symmetric group $ S_n $.

References

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