Automatic supermartingales acting on sequences

TL;DR

This paper introduces automatic supermartingales realized as automatic functions, linking their properties to the notion of randomness in infinite binary sequences, and showing their equivalence to disjunctive sequences.

Contribution

It constructs automatic supermartingales using automatic capital groups and studies their properties, connecting them to randomness and disjunctive sequences.

Findings

Automatic supermartingales are realized as automatic functions.

The class of random sequences coincides with disjunctive sequences.

Properties of automatic supermartingales are systematically studied.

Abstract

This paper describes a construction of supermartingales realized as automatic functions. A capital of supermartingales is represented using automatic capital groups~(ACG). Properties of these automatic supermartingales are then studied. Automatic supermartingales induce a notion of random infinite binary sequence. We show that the class of random sequences coincide with that of disjunctive sequences.