How powerful are integer-valued martingales?

TL;DR

This paper explores the effectiveness of integer-valued martingales in the context of algorithmic randomness, comparing their power to that of real-valued martingales in detecting randomness.

Contribution

It introduces and analyzes a restricted model of martingales with integer bets, examining its implications for the concept of computable randomness.

Findings

Integer-valued martingales form a distinct class of predictors.

They are less powerful than real-valued martingales in detecting non-random sequences.

The class of sequences random with respect to integer-valued martingales differs from classical notions.

Abstract

In the theory of algorithmic randomness, one of the central notions is that of computable randomness. An infinite binary sequence X is computably random if no recursive martingale (strategy) can win an infinite amount of money by betting on the values of the bits of X. In the classical model, the martingales considered are real-valued, that is, the bets made by the martingale can be arbitrary real numbers. In this paper, we investigate a more restricted model, where only integer-valued martingales are considered, and we study the class of random sequences induced by this model.