A new approach to strong embeddings

Abstract

We revisit strong approximation theory from a new perspective, culminating in a proof of the Komlos–Major–Tusnady 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.