A stochastic maximal inequality, strict countability, and infinite-dimensional martingales

TL;DR

This paper introduces a new stochastic maximal inequality and the concept of strict countability to establish weak convergence results for separable random fields and martingales, providing novel insights into empirical process theory.

Contribution

It presents a novel method based on stochastic maximal inequalities and strict countability, offering new weak convergence theorems and results for empirical processes and i.i.d. sequences.

Findings

New stochastic maximal inequality derived using integration by parts

Weak convergence theorems for separable random fields of martingales

A new Donsker theorem and moment bounds for empirical processes

Abstract

As an alternative to the well-known methods of "chaining" and "bracketing" that have been developed in the study of random fields, a new method, which is based on a {\em stochastic maximal inequality} derived by using the formula for integration by parts and on a new concept named {\em strict countability}, is presented. The main results are some weakconvergence theorems for sequences of separable random fields of discrete-time martingales under the uniform topology with the help also of entropy methods. As special cases, some new results for i.i.d.\ random sequences, including a new Donsker theorem and a moment bound for suprema of empirical processes indexed by classes of sets or functions, are obtained.