Dvoretzky type theorems for subgaussian coordinate projections

Given a class of functions $F$ on a probability space $(\mathrm{\Omega },\mu )$, we study the structure of a typical coordinate projection of the class, defined by $\{(f(X_i){)}_{i=1}^N:f\in F\}$, where $X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_N$ are independent, selected according to $\mu$. 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.