The suprema of selector processes with the application to positive infinitely divisible processes

TL;DR

The paper tackles the problem of bounding the expected supremum of nonnegative selector, empirical, and infinitely divisible processes by developing a threshold-based witnessing framework built around fragment ideas and $p$-smallness. It shows that, for not-$p$-small index families, one can obtain explicit lower bounds on the expected supremum, and then constructs small covers of the large-supremum event using witnesses, enabling a decomposition of complex processes into a gamma-functional (chaining) component and a positive-process component. By extending Park-Pham’s results to positive infinitely divisible processes via Lévy measures and Poisson point processes, the authors provide a unified approach that yields small-cover bounds in selector, infinitely divisible, and empirical settings. The framework has potential applications in bounding operator norms of random matrices and related high-dimensional probabilistic problems, and it paves the way for further generalizations beyond the i.i.d. case.

Abstract

We provide an alternative proof of the recent result by Park and Pham (2022) on the expected suprema of positive selector and empirical processes. We extend it to positive infinitely divisible processes.

Theorems & Definitions (58)

• Theorem 2.1
• Remark 2.2
• Theorem 2.3
• Theorem 3.1
• Remark 3.2
• Definition 3.3
• Lemma 3.4: Key Lemma
• proof : Proof of Theorem \ref{['main2']}
• Remark 3.5
• Lemma 3.6