The Law of the Iterated Logarithm for Algorithmically Random Brownian Motion

Authors:
Bjørn Kjos-Hanssen;Anil Nerode
Affiliations:
Department of Mathematics, Cornell University, Ithaca, NY 14853, U.S.A.;Department of Mathematics, Cornell University, Ithaca, NY 14853, U.S.A.
Venue:
LFCS '07 Proceedings of the international symposium on Logical Foundations of Computer Science
Year:
2007

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Abstract

Algorithmic randomness is most often studied in the setting of the fair-coin measure on the Cantor space, or equivalently Lebesgue measure on the unit interval. It has also been considered for the Wiener measure on the space of continuous functions. Answering a question of Fouché, we show that Khintchine's law of the iterated logarithm holds at almost all points for each Martin-Löf random path of Brownian motion. In the terminology of Fouché, these are the complex oscillations. The main new idea of the proof is to take advantage of the Wiener-Carathéodory measure algebra isomorphism theorem.