The optimal order for the \(p\)-th moment of sums of independent random variables with respect to symmetric norms and related combinatorial estimates (Q2499042)

Based on the author's summary: ``For \(n\) independent random variables \(f_{1}, \dots, f_{n}\) and a symmetric norm \(| | \;| | _{X}\) on \(\mathbb R^{n}\), we show that for \(1\leq p < \infty\) \[ \begin{aligned}\frac{1}{2+4\sqrt{2}}\Biggl( (n \int_0^{\frac{1}{n}} &h^* (s)^p ds)^{\frac{1}{p}} + \left\| \sum_{i=1}^{n}(n \int_{\frac{i-1}{n}}^{\frac{i}{n}}h^*(s)ds)e_i\right\| _X\Biggr)\\ &\leq \Biggl(\int \left\| \sum_{i=1}^n f_ie_i\right\| _X^p d\mu\Biggr)^{\frac{1}{p}}\\ &\leq \frac{c_0p}{1+\ln p}\Biggl( (n \int_0^{\frac{1}{n}} h^* (s)^p ds)^{\frac{1}{p}} + \left\| \sum_{i=1}^{n}(n \int_{\frac{i-1}{n}}^{\frac{i}{n}}h^*(s)ds)e_i\right\| _X\Biggr).\end{aligned} \] Here \[ h(t,\omega) = \sum_{i=1}^n 1_{[\frac{i-1}{n},\frac{i}{n})}(t)f_i(\omega) \] is the disjoint sum of the \(f_{i}'s\) and \(h^*\) is the non-increasing rearrangement. Similar results (where \(L_{p}\) is replaced by a more general rearrangement invariant function space) were obtained first by \textit{Y. Gordon, A. Litvak, C. Schütt} and \textit{E. Werner} [Ann. Probab. 30, 1833--1853 (2002; Zbl 1016.60008)] for Orlicz spaces \(X\) and independently by \textit{S. Montgomery-Smith} [Isr. J. Math. 131, 51--60 (2002; Zbl 1010.60041)] for general \(X\) but without an explicit analysis of the order of growth for the constant in the upper estimate. The order \(\frac{p}{1+\ln p}\) is optimal and obtained from combinatorial estimates for doubly stochastic matrices. The result extends to Lorentz-norms \(l_{f, q}\) on \(\mathbb R^{n}\) under mild assumptions on \(f\). We give applications to the theory of noncommutative \(L_{p}\) spaces.'' In Theorem 1.1 cited in this paper the non-decreasing rearrangement of the matrix actually consists of \(n^{2}\) elements \(\alpha_{1}^{*}, \dots, \alpha_{n^{2}}^{*}\) but only the first \(n\) of them play a role in this version of the Kwapień-Schütt estimate.