On weakly bounded empirical processes

Let $F$ be a class of functions on a probability space $(\mathrm{\Omega },\mu )$ and let $X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_k$ be independent random variables distributed according to $\mu$. We establish high probability tail estimates of the form $\sup_{f \in F} |{i : |f(X_i)| \geq t }$usinganaturalparameterassociatedwith$F$. We use this result to analyze weakly bounded empirical processes indexed by $F$ and processes of the form $Z_f=\mathrm{\mid }{k}^{-1}{\sum }_{i=1}^k\mathrm{\mid }f{\mathrm{\mid }}^p(X_i)-\text{E}\mathrm{\mid }f{\mathrm{\mid }}^p\mathrm{\mid }$ for $p>1$. We also present some geometric applications of this approach, based on properties of the random operator $\Gamma=k^{-1/2}\sum_{i=1}^k \inr{X_i,\cdot}e_i$,wherethe$(X_i)_{i=1}^k$ are sampled according to an isotropic, log-concave measure on ${\mathbb{R}}^n$.