Randomness and initial segment complexity for probability measures

TL;DR

This paper explores a weak form of algorithmic randomness for probability measures on Cantor space, linking initial segment complexity growth to measure randomness properties and examining triviality and ergodic measures.

Contribution

It introduces a new weak randomness notion for measures, connects it to initial segment complexity growth, and analyzes its relation to classical randomness and triviality notions.

Findings

Maximal growth of complexity implies weak randomness.

Full Martin-Löf randomness of a measure implies ML absolute continuity.

The converse of this implication does not hold due to atoms in measures.

Abstract

We study algorithmic randomness properties for probability measures on Cantor space. We say that a measure $\mu$ on the space of infinite bit sequences is ML absolutely continuous if the non-ML-random bit sequences form a null set with respect to~$\mu$. We think of this as a weak randomness notion for measures. We begin with examples, and provide a robustness property related to Solovay tests. Our main work connects our weak randomness notion to the growth of the initial segment complexity for measures~$\mu$; the latter is defined as a $\mu$-average over the complexity of strings of the same length. We show that a maximal growth implies our weak randomness property, but also that both implications of the Levin-Schnorr theorem fail. We discuss $C$-triviality and $K$-triviality for measures and relate these two notions with each other. Here triviality means that the growth of initial segment complexity is as slow as possible. We show that full Martin-L\"of randomness of a measure implies ML absolute continuity; the converse fails because only the latter property is compatible with having atoms. In a final section we consider weak randomness relative to a general ergodic computable measure. We seek appropriate effective versions of the Shannon-McMillan-Breiman theorem and the Brudno theorem where the bit sequences are replaced by measures. We conclude with several open questions.