On weakly bounded empirical processes

Let F be a class of functions on a probability space (Ω, μ) and let X1, ..., Xk be independent random variables distributed according to μ. We establish an upper bound that holds with high probability on supf∈F |{i : |f(Xi)| ≥ t} for every t > 0, and that depends on a natural geometric parameter associated with F . We use this result to analyze the supremum of empirical processes of the form Zf = ∣∣k−1 ki=1 |f |(Xi)− E|f | ∣∣ for p > 1 using the geometry of F . We also present some geometric applications of this approach, based on properties of the random operator Γ = k−1/2 ∑k i=1 〈 Xi, · 〉 ei, where (Xi) k i=1 are sampled according to an isotropic, log-concave measure on R.