A sharpening of Tusnády'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 $X_m$. Tusnády's inequality states that $$\mid {\Psi }_m(Y_m)-Y_m\mid \le \frac{Y_m^2}{m}+1.$$ Here we propose a sharpened version of this inequality.

7 pages and 2 figures in pdf files

References in corpus (2)

• Tusnady's inequality revisited
• Improvement of two Hungarian bivariate theorems