On weakly bounded empirical processes

Abstract

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

References in corpus (1)

• Reconstruction and subgaussian operators

Cited by in corpus (1)

• Discussion of ``2004 IMS Medallion Lecture: Local Rademacher complexities and oracle inequalities in risk minimization'' by V. Koltchinskii