A Note on Distribution Free Symmetrization Inequalities

Let $X,Y$ be two independent identically distributed (i.i.d.) random variables taking values from a separable Banach space $($\mathcal{X}$, |\cdot|)$\mathrm{.}Giventwomeasurablesubsets$F, K\subseteq\cal{X}$, 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 $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\mathrm{\infty }}$ norm. Our approach for both problems extends techniques developed by Schultze and Weizs"acher (2007).