Maximum Inequalities for Rearrangements of Summands and Assignments of Signs

The interrelation between signs and permutations in maximum inequalities is studied in this paper. The relationship is based on a lemma that reduces a rearrangement problem to a problem of choosing signs. It helps simplify proofs and find new facts and general settings. The following inequality is one of the main results of the paper: Let $x_1,\ldots , x_n$, $\sum_{k=1}^n x_k =0,$ be a collection of elements of a normed space $X$. Then for any collection of signs $\theta = (\theta_1 , \ldots , \theta_n)$ and any $t>0$, $ {\bf P} \{\pi : \max_{1\le k \le n}\Vert\sum_1^k x_{\pi (i)}\Vert > t\} \le C{\bf P}\{\pi : \max_{1\le k \le n}\Vert \sum_1^k x_{\pi (i)}\theta_i\Vert> \frac {t}{C}\}, $ where $\pi\in \Pi_n ,$ $\Pi_n$ is the group of all permutations of $\{1,\ldots ,n\}$, $P$ is the uniform distribution on it, and $C$ is an absolute constant. The inequality is unimprovable (the inverse inequality also holds with some other constant); it generalizes well-known results due to Garsia; Maurey, and Pisier; Kas...