A stochastic maximal inequality, strict countability, and related topics

TL;DR

This paper introduces a novel stochastic maximal inequality and the concept of strict countability, providing new weak convergence results for random fields and applications to empirical processes and semiparametric statistical estimation.

Contribution

It presents a new method based on stochastic maximal inequalities and strict countability, offering alternative approaches to chaining and bracketing in random field analysis.

Findings

New weak convergence theorems for random fields of martingales

A novel Donsker theorem for i.i.d. sequences

Moment bounds for empirical process suprema

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 stochastic maximal inequality derived by using It\^o's formula and on a new concept named strict countability, is presented. The main results are some weak convergence theorems for sequences of separable random fields of locally square-integrable 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. An application to statistical estimation in semiparametric models is presented with an illustration to construct adaptive estimators in Cox's regression model.