Estimating averages of order statistics of bivariate functions

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.