The Law of the Iterated Logarithm for Algorithmically Random Brownian Motion

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.