An alternative construction of the strong embedding for the simple random walk

We give a new proof of the Komlos-Major-Tusnady embedding theorem for the simple random walk. The only external tool that we use is the Schauder-Tychonoff fixed point theorem for locally convex spaces. Besides that, the proof is almost entirely based on a series of soft arguments and easy inequalities, and no hard computations (implicit or explicit) are involved. This provides the first genuine alternative to the quantile transform and the Hungarian construction.