A sharpening of Tusn\'ady's inequality

TL;DR

This paper presents a refined version of Tusnady's inequality, providing a tighter bound on the difference between a transformed normal variable and a sum of i.i.d. Rademacher variables, enhancing probabilistic approximation accuracy.

Contribution

The authors introduce a sharper inequality that improves the bounds of Tusnady's original inequality for better probabilistic approximations.

Findings

The new inequality offers a tighter bound on the difference between the transformed normal and the sum.

It enhances the accuracy of probabilistic coupling methods.

The result has potential applications in probability theory and statistical approximation techniques.

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.