Let \(X,X^{\prime }\) be independent identically distributed real random variables on a probability space, let \(1\le q\le \infty \), assume \(X\in L^q\) and \(\mathbb {E}X=0\), and let \(a\in \mathbb {R}\). Then \(\| a+X\| _q\le \| a+X-X^{\prime }\| _q\), with the essential supremum at \(q=\infty \).

Let \(X\) be a centered real random variable on a probability space, let \(1\le q\le \infty \), and assume \(X\in L^q\). If \(r\) is an independent measurable uniform sign in \(\{ -1,1\} \), then for every \(a\in \mathbb {R}\), \(\| a+X/2\| _q\le \| a+rX\| _q\), including the essential-supremum norm.