Algorithmic information theory and martingales

TL;DR

This paper reviews the historical development of algorithmic information theory and martingales, highlighting their roles in understanding individual randomness and foundational aspects of probability theory.

Contribution

It provides a comprehensive overview of the evolution of the concepts of martingales and algorithmic information theory from their origins to their foundational applications.

Findings

Historical analysis of martingales and randomness concepts

Development of algorithmic information theory in the 1960s and 1970s

Remarks on applications in the foundations of probability

Abstract

The notion of an individual random sequence goes back to von Mises. We describe the evolution of this notion, especially the use of martingales (suggested by Ville), and the development of algorithmic information theory in 1960s and 1970s (Solomonov, Kolmogorov, Martin-Lof, Levin, Chaitin, Schnorr and others). We conclude with some remarks about the use of the algorithmic information theory in the foundations of probability theory.