Nonuniform Berry–Esseen bound for character ratios

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 $S_n$.