The Dual Majorizing Measure Theorem for Canonical Processes

TL;DR

This paper presents a dual formulation of Latala's majorizing measure theorem for canonical processes with log-concave tails, introducing parameterized separation trees and a polynomial-time approximation algorithm.

Contribution

It introduces a dual, separated-tree formulation and a deterministic polynomial-time algorithm for approximating the expected supremum of canonical processes.

Findings

Expected supremum is equivalent to a tree functional up to universal constants

Develops a pointwise growth condition inspired by the contraction principle

Provides a polynomial-time algorithm for finite index sets

Abstract

We give a dual, separated-tree formulation of Latala's majorizing measure theorem for canonical processes with log-concave tails. Under the same assumptions as in Latala's characterization, we introduce parameterized separation trees and prove that the expected supremum is equivalent, up to universal constants, to the corresponding tree functional. We also develop a pointwise growth condition, inspired by the contraction principle, which leads to a deterministic polynomial-time algorithm for approximating the expected supremum when the index set is finite.