Kjos-Hanssen, Bjørn; Nguyen, Paul Kim Long V.; Rute, Jason Algorithmic randomness for Doob’s martingale convergence theorem in continuous time. (English) Zbl 1325.03051 Log. Methods Comput. Sci. 10, No. 4, Paper No. 12, 35 p. (2014). Summary: 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. MSC: 03D32 Algorithmic randomness and dimension 60J65 Brownian motion 60G44 Martingales with continuous parameter Keywords:Doob’s martingale convergence theorem; algorithmic randomness; computability theory; Schnorr randomness; computable randomness; Brownian motion PDF BibTeX XML Cite \textit{B. Kjos-Hanssen} et al., Log. Methods Comput. Sci. 10, No. 4, Paper No. 12, 35 p. (2014; Zbl 1325.03051) Full Text: DOI arXiv