## Abstract

Let F be a class of functions on a probability space (Ω, μ) and let X _{1},...,X _{k} be independent random variables distributed according to μ. We establish an upper bound that holds with high probability on {\rm sup}_{f \in F} |\{i : |f(X_i)| \geq 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 Z_{f} = |k_{-1}∑_{i=1}^{k}|f|^{p}(X_{i}) - double-struck E sign|f|^{p}| 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}\sum_{i=1}X _{i} , •e _{i} , where (X_i)_{i=1}k are sampled according to an isotropic, log-concave measure on {\mathbb R.

Original language | English |
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Pages (from-to) | 293-314 |

Number of pages | 22 |

Journal | Mathematische Annalen |

Volume | 340 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 2008 |