Chobanyan, Sergei; Levental, Shlomo; Salehi, Habib A distribution maximum inequality for rearrangements of summands. (English) Zbl 1259.60023 Bull. Georgian Natl. Acad. Sci. (N.S.) 5, No. 3, 25-30 (2011). Let \(x_1,\dotsc,x_n\) be elements of a normed space \((X,\left\| \cdot\right\|\)) with \(\sum_{i=1}^nx_i=0\), and let \(\Pi_n\) be the set of all permutations of \((1,\dotsc,n)\). In this paper, it is shown that there exists a constant \(C>0\) such that \[ \left |\left\{ \pi\in\Pi_n:\,\max_{1\leq k\leq n} \left\| \sum_{i=1}^k x_{\pi(i)}\right\| > t\right\}\right | \leq C \left |\left\{ \pi\in\Pi_n:\,\max_{1\leq k\leq n} \left\| \sum_{i=1}^k \vartheta_i x_{\pi(i)}\right\| > t/C\right\}\right | \] for any collection \(\vartheta_1,\dotsc,\vartheta_n\in\{-1,1\}\) of signs and any \(t>0\), where \(|A|\) denotes the number of elements of a finite set \(A\). Reviewer: Michael Falk (Würzburg) Cited in 2 Documents MSC: 60E15 Inequalities; stochastic orderings 60B11 Probability theory on linear topological spaces Keywords:permutations of summands; maximum inequality for rearrangements of summands PDFBibTeX XMLCite \textit{S. Chobanyan} et al., Bull. Georgian Natl. Acad. Sci. (N.S.) 5, No. 3, 25--30 (2011; Zbl 1259.60023)