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arXiv · 1411.0186

Algorithmic randomness for Doob's martingale convergence theorem in continuous time

Bjørn Kjos-Hanssen·Paul Kim Long V. Nguyen·Jason Rute
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Abstract

We study Doob's martingale convergence theorem for computable continuous time martingales on Brownian motion, in the context of algorithmic randomness. A characterization of the class of sample points for which the theorem holds is given. Such points are given the name of Doob random points. It is shown that a point is Doob random if its tail is computably random in a certain sense. Moreover, Doob randomness is strictly weaker than computable randomness and is incomparable with Schnorr randomness.

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Bjørn Kjos-Hanssen, Paul Kim Long V. Nguyen, Jason Rute. 2014-11-02. Algorithmic randomness for Doob's martingale convergence theorem in continuous time. https://doi.org/10.2168/lmcs-10(4%3A12)2014

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