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