Algorithmic randomness for Doob's martingale convergence theorem in continuous time
Bjørn Kjos-Hanssen, Paul Kim Long V. Nguyen, Jason Rute · Logical Methods in Computer Science · 2014
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.