From oracle maximal inequalities to martingale random fields via finite approximation from below

TL;DR

This paper introduces a new method for bounding the expected supremum of martingale random fields, extending classical inequalities and applying finite approximation techniques to derive convergence results and moment bounds.

Contribution

It develops an oracle maximal inequality for finite classes of submartingales and generalizes Lenglart's inequality to higher dimensions and infinite-dimensional settings.

Findings

Established a sharp upper bound for the supremum of martingale random fields.

Derived a generalized Lenglart's inequality for multi-dimensional martingales.

Proved new weak convergence theorems and moment bounds for empirical processes.

Abstract

A novel approach is proposed to establish a sharp upper bound on the expected supremum of a separable martingale random field, serving as an alternative to classical universal chaining-based methods. The proposed approach begins by deriving a new "oracle maximal inequality" for a finite class of submartingales. This is achieved via integration by parts rather than a simplistic application of the triangle inequality. Consequently, we obtain a generalization of Lenglart's inequality for discrete-time martingales, extending it from the one-dimensional case to finite-dimensional settings, and further to certain infinite-dimensional cases through a "finite approximation device". The primary applications include several weak convergence theorems for sequences of separable martingale random fields under the uniform topology. In particular, new results are established for i.i.d. sequences, including a necessary and sufficient condition for a countable class of functions to possess the Donsker property. Additionally, we provide new moment bounds for the supremum of empirical processes indexed by classes of sets or functions.