A Note on Distribution Free Symmetrization Inequalities

Abstract

Let $X,Y$ be two independent identically distributed (i.i.d.) random variables taking values from a separable Banach space $(\mathcal{X},\Vert \cdot \Vert )$. Given two measurable subsets $F,K\subseteq \mathcal{X}$, we established distribution free comparison inequalities between $\mathbb{P}(X\pm Y\in F)$ and $\mathbb{P}(X-Y\in K)$. These estimates are optimal for real random variables as well as when $\mathcal{X}={\mathbb{R}}^d$ is equipped with the $\Vert \cdot {\Vert }_{\infty }$ norm. Our approach for both problems extends techniques developed by Schultze and Weizsächer (2007).