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 $S_n$.

In 1962, Tallis introduced the generalized multinomial model for a sequence of random variables $(X_j)$. This model generalizes the independent case to a dependent structure resembling equicorrelation, which is meaningful in practice because it captures scenarios in which several risks or variables are jointly driven by a common underlying factor, causing them to move together with comparable strength. Within this model, we denote $W=X_1+X_2+\cdots +X_N$ as a random sum with a random index $N$, and $Z$ as the standard normal random variable. Our goal is to establish a non-uniform bound for the stop-loss distance, $|E(W-k{)}^{+}-E(Z-k{)}^{+}|$. In this work, in addition to biasing ideas, we apply Stein's method together with an appropriately chosen test function, which allows us to effectively use the mean value theorem. Moreover, our results are illustrated through a realistic application involving the quantity $E(W-k{)}^{+}$, which arises naturally in many financial and insurance settings, as it represents the expected excess of a loss or payoff above a specified threshold $k$.