Kolmogorov complexity and strong approximation of Brownian motion

Kjos-Hanssen, Bjørn; Szabados, Tamás

doi:10.1090/S0002-9939-2011-10741-X

Kolmogorov complexity and strong approximation of Brownian motion
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by Bjørn Kjos-Hanssen and Tamás Szabados PDF

Proc. Amer. Math. Soc. 139 (2011), 3307-3316 Request permission

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-Löf random paths of Brownian motion have such an incompressible close approximant. This strengthens a result of Asarin, who obtained instead the bound $O(n^{-1/6} \log n)$. The result cannot be improved to $o(n^{-1/2}{\sqrt {\log n}})$.

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