Uniform local probability approximations: Improvements on Berry-Esseen (Q1897192)

Let \((X_i)\) be i.i.d. mean-zero, bounded r.v.s, \(S_n= X_1+\cdots+ X_n\). The authors prove under certain conditions results of the type \[ P(S_n\in \Gamma)= P(S_n/\sqrt{\text{var}(S_n)}\in \Gamma/\sqrt{\text{var}(S_n)})\approx |\Gamma|/ \sqrt{\text{var}(S_n)}, \] where \(|\cdot|\) denotes Lebesgue measure. This implies, in particular, that, whenever \(\Gamma\) intersects the interval between the \(\varepsilon\)th and the \((1- \varepsilon)\)th quantile of \(S_n\), \(P(S_n\in \Gamma)\) is proportional to the length of the interval over the standard deviation.