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

TL;DR

This paper explores the concept of Doob randomness in the context of computable continuous-time martingales on Brownian motion, characterizing points where Doob's martingale convergence theorem applies and comparing different notions of randomness.

Contribution

It introduces Doob random points, characterizes them, and compares Doob randomness with computable and Schnorr randomness in continuous time.

Findings

Doob random points are characterized by tail computable randomness.

Doob randomness is weaker than computable randomness.

Doob randomness is incomparable with Schnorr randomness.

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.