On weakly bounded empirical processes

Shahar Mendelson*

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    13 Citations (Scopus)

    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 Zf = |k-1i=1k|f|p(Xi) - 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 languageEnglish
    Pages (from-to)293-314
    Number of pages22
    JournalMathematische Annalen
    Volume340
    Issue number2
    DOIs
    Publication statusPublished - Feb 2008

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