Dvoretzky type theorems for subgaussian coordinate projections

Abstract

Given a class of functions $F$ on a probability space $(Ω,μ)$, we study the structure of a typical coordinate projection of the class, defined by $\{(f(X_i))_{i=1}^N : f \in F\}$, where $X_1,...,X_N$ are independent, selected according to $μ$. This notion of projection generalizes the standard linear random projection used in Asymptotic Geometric Analysis. We show that when $F$ is a subgaussian class of functions, a typical coordinate projection satisfies a Dvoretzky type theorem.