A sharpening of Tusn\'ady's inequality

Abstract

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 same first two moments as that of $X_m.$ There is a uniquely determined function $\Psi_m$ such that the distribution of $\Psi_m(Y_m)$ equals to the distribution of $X_m$. Tusn\'ady's inequality states that $$ \mid \Psi_m(Y_m) - Y_m \mid \leq \frac{Y_m^2}{m}+1.$$ Here we propose a sharpened version of this inequality.