Bounding suprema of canonical processes via convex hull

TL;DR

This paper introduces a new approach to bounding the supremum of canonical processes by leveraging convex hull inclusion, broadening applicability to various index sets like ellipsoids with weaker regularity assumptions.

Contribution

It extends the method of bounding suprema to more general index sets, including ellipsoids, with less restrictive regularity conditions than previously required.

Findings

Upper bounds are straightforward to obtain.

Reverse bounds are now applicable to a wider class of processes.

Applicable to arbitrary ellipsoids with weakened assumptions.

Abstract

We discuss the method of bounding suprema of canonical processes based on the inclusion of their index set into a convex hull of a well-controlled set of points. While the upper bound is immediate, the reverse estimate was established to date only for a narrow class of regular stochastic processes. We show that for specific index sets, including arbitrary ellipsoids, regularity assumptions may be substantially weakened.