Kolmogorov complexity and strong approximation of Brownian motion

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 n -step walk is within a uniform distance O ( n − 1 / 2 log ⁡ n ) O(n^{-1/2}\log n) of the Brownian path for all but finitely many positive integers n n . Almost surely this n n -step walk will be incompressible in the sense of Kolmogorov complexity, and all Martin-Löf random paths of Brownian motion have such an incompressible close approximant. This strengthens a result of Asarin, who obtained instead the bound O ( n − 1 / 6 log ⁡ n ) O(n^{-1/6} \log n) . The result cannot be improved to o ( n − 1 / 2 log ⁡ n ) o(n^{-1/2}{\sqrt {\log n}}) .