On Instability of the Ergodic Limit Theorems with Respect to Small Violations of Algorithmic Randomness

TL;DR

This paper investigates how small deviations from perfect algorithmic randomness can cause instability in fundamental ergodic theorems and related asymptotic laws, including the Shannon--McMillan--Breiman theorem and universal compression schemes.

Contribution

It introduces the concept of instability in ergodic theorems due to minor violations of algorithmic randomness, highlighting their impact on classical results.

Findings

Ergodic theorems are sensitive to small randomness violations.

Universal compression schemes are affected by these instabilities.

The study reveals limits of robustness in asymptotic laws.

Abstract

An instability property of the Birkhoff's ergodic theorem and related asymptotic laws with respect to small violations of algorithmic randomness is studied. The Shannon--McMillan--Breiman theorem and all universal compression schemes are also among them.