  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 $F, K\subseteq\cal{X}$, 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 $\|\cdot\|_\infty$ norm. Our approach for both problems extends techniques
developed by Schultze and Weizs\"acher (2007).
