Kolmogorov complexity and strong approximation of Brownian motion

TL;DR

This paper demonstrates that Brownian motion can be strongly approximated by simple random walks with high Kolmogorov complexity, improving previous bounds and establishing the incompressibility of such approximations.

Contribution

The paper proves a stronger almost sure uniform approximation of Brownian motion by simple random walks with high Kolmogorov complexity, refining earlier bounds.

Findings

Almost sure uniform approximation within O(n^{-1/2} log n)

Incompressibility of the approximating random walks

Bound cannot be improved to o(n^{-1/2} sqrt{log n})

Abstract

Brownian motion and scaled and interpolated simple random walk can be jointly embedded in a probability space in such a way that almost surely the $n$-step walk is within a uniform distance $O(n^{-1/2}\log n)$ of the Brownian path for all but finitely many positive integers $n$. Almost surely this $n$-step walk will be incompressible in the sense of Kolmogorov complexity, and all {Martin-L\"of random} paths of Brownian motion have such an incompressible close approximant. This strengthens a result of Asarin, who obtained the bound $O(n^{-1/6} \log n)$. The result cannot be improved to $o(n^{-1/2}{\sqrt{\log n}})$.