Estimating averages of order statistics of bivariate functions

TL;DR

This paper develops uniform estimates for the expected averages of order statistics of bivariate functions and sequences of independent random variables, with applications to Orlicz norms and embedding properties of matrices.

Contribution

It introduces new uniform estimates for order statistics and provides a minimal probability space for embedding Orlicz spaces into $ ext{l}_1$ spaces.

Findings

Uniform estimates for order statistics in terms of largest values.

Application to Orlicz norms of independent random variables.

Embedding of Orlicz spaces into $ ext{l}_1^{cn^3}$.

Abstract

We prove uniform estimates for the expected value of averages of order statistics of bivariate functions in terms of their largest values by a direct analysis. As an application, uniform estimates for the expected value of averages of order statistics of sequences of independent random variables in terms of Orlicz norms are obtained. In the case where the bivariate functions are matrices, we provide a "minimal" probability space which allows us to $C$-embed certain Orlicz spaces $\ell_M^n$ into $\ell_1^{cn^3}$, $c,C>0$ being absolute constants.