Dimensionality Reduction in Extremal Problems for Moments of Linear Combinations of Vectors with Random Coefficients

Abstract

It is shown that

$$\sup \left\{ {E\varphi \left( {{{{\left\| {\sum\limits_{{i = 1}}^{n} {{{\xi }_{i}}{{x}_{i}}} } \right\|}}^{2}}} \right):{{x}_{i}} \in H, \sum\limits_{{i = 1}}^{n} {\parallel {{x}_{i}}{{\parallel }^{2}} = {{B}^{2}}} } \right\}$$

does not depend on dim H ≥ 1, where $(H,\parallel \cdot \parallel )$ is a Hilbert space, φ is any convex function, and ξ ₁, …, ξ _n are any (real-valued) random variables. An immediate corollary is the following vector extension of the Whittle—Haagerup inequality: let ε ₁, …, ε _n be independent Rademacher random variables, and let x ₁, …, x _n be vectors in H; then

$$\begin{array}{*{20}{c}} {E{{{\left\| {\sum\limits_{{i = 1}}^{n} {{{\varepsilon }_{i}}{{x}_{i}}} } \right\|}}^{p}} \leqslant E|\nu {{|}^{p}}{{{\left( {\sum\limits_{{i = 1}}^{n} {\parallel {{x}_{i}}{{\parallel }^{2}}} } \right)}}^{{p/2}}}} & {\forall p \geqslant 2,} \\ \end{array}$$

where v ~ N(0, 1). Dimensionality reduction in the case when all the lengths $\parallel {{x}_{i}}\parallel$ are fixed is also considered. Open problems are stated.