A new approach to strong embeddings

TL;DR

This paper introduces a novel approach to strong approximation theory, providing a new proof of the Komlós-Major-Tusnády embedding theorem for simple random walks using techniques inspired by Stein's method, with potential broader applications.

Contribution

It presents a new technique based on Stein's method for strong approximation, extending the applicability beyond sums of independent variables.

Findings

New proof of the Komlós-Major-Tusnády embedding theorem

Technique applicable to various settings where Stein's method is effective

Potential to extend strong approximation methods beyond traditional sums

Abstract

We revisit strong approximation theory from a new perspective, culminating in a proof of the Koml\'os-Major-Tusn\'ady embedding theorem for the simple random walk. The proof is almost entirely based on a series of soft arguments and easy inequalities. The new technique, inspired by Stein's method of normal approximation, is applicable to any setting where Stein's method works. In particular, one can hope to take it beyond sums of independent random variables.