A Peculiar Proof of the Martingale Convergence Theorem

TL;DR

This paper presents a unique proof of the Martingale Convergence Theorem by leveraging embedding techniques related to Brownian motion, offering a novel perspective on classical probabilistic convergence results.

Contribution

It introduces a new proof method for the Martingale Convergence Theorem based on embedding martingales into Brownian motion sample paths, building on Dubins and Monroe's work.

Findings

Provides a novel proof of the Martingale Convergence Theorem

Utilizes embedding of martingales into Brownian motion

Connects discrete-time martingales with continuous-time processes

Abstract

We prove the Martingale Convergence Theorem by using the work of L. Dubins and I. Monroe about embedding a given discrete-time martingale in the sample paths of a Brownian motion.