Random embedding of $${\ell_p^n}$$ into $${\ell_r^N}$$

Abstract

For any 0 n, we give an explicit definition of a random operator \({S : \ell_p^n \to \mathbb{R}^N}\) such that for every 0 0 is a real number depending only on p and r. One of the main tools that we develop is a new type of multidimensional Esseen inequality for studying small ball probabilities.