Ergodic theorems for algorithmically random points

TL;DR

This survey explores how the theory of algorithmic randomness, particularly Martin-Lof randomness, can be used to formulate and understand ergodic theorems in a constructive, pointwise manner, bridging probability theory and algorithmic information.

Contribution

It demonstrates that the Birkhoff ergodic theorem, traditionally non-constructive, can be interpreted in a weaker constructive sense using Martin-Lof random points.

Findings

Birkhoff ergodic theorem is non-constructive classically

It can be made constructive via Martin-Lof randomness

Establishes degrees of constructivity for probabilistic laws

Abstract

This paper is a survey of applications of the theory of algorithmic randomness to ergodic theory. We establish various degrees of constructivity for asymptotic laws of probability theory. In the framework of the Kolmogorov approach to the substantiation of the probability theory and information theory on the base of the theory of algorithms, we formulate probabilistic laws, i.e. statements which hold almost surely, in a pointwise form, i.e., for Martin-Lof random points. It is shown in this paper that the main statement of ergodic theory - Birkhoff ergodic theorem, is non-constructive in the strong (classical) sense, but it is constructive in some weaker sense - in terms of Martin-Lof randomness.