The extended Ville's inequality for nonintegrable nonnegative supermartingales

TL;DR

This paper extends Ville's inequality to nonnegative supermartingales without the need for integrability, broadening its applicability in sequential statistics and nonparametric inference.

Contribution

It introduces an extended Ville's inequality for nonintegrable supermartingales and develops a method of mixtures applicable to sigma-finite cases, advancing sequential analysis.

Findings

Derived a maximal inequality for nonintegrable supermartingales.

Extended the method of mixtures to sigma-finite supermartingales.

Implications for nonparametric confidence sequences and e-processes.

Abstract

Following the initial work by Robbins, we rigorously present an extended theory of nonnegative supermartingales, requiring neither integrability nor finiteness. In particular, we derive a key maximal inequality foreshadowed by Robbins, which we call the extended Ville's inequality, that strengthens the classical Ville's inequality (for integrable nonnegative supermartingales), and also applies to our nonintegrable setting. We derive an extension of the method of mixtures, which applies to $\sigma$-finite mixtures of our extended nonnegative supermartingales. We present some implications of our theory for sequential statistics, such as the use of improper mixtures (priors) in deriving nonparametric confidence sequences and (extended) e-processes.