A sharpening of Tusnády's inequality

Jenő Reiczigel, Lídia Rejtő, Gábor Tusnády

Let ~$\veps_1, ..., \veps_m$ be i.i.d. random variables with $$P(\veps_i=1)= P(\veps_i= -1)=1/2,$$ and $X_m = \sum_{i=1}^m \veps_i.$ Let $Y_m$ be a normal random variable with the…