Algorithmic randomness and monotone complexity on product space

TL;DR

This paper investigates algorithmic randomness and monotone complexity on product spaces of infinite binary sequences, extending Lambalgen's theorem without the need for a uniform computability assumption and exploring related statistical and complexity concepts.

Contribution

It generalizes Lambalgen's theorem for correlated probability without the uniform computability assumption and studies various aspects of randomness and complexity on product spaces.

Findings

Lambalgen's theorem is proven without the uniform computability assumption.

Analysis of monotone complexity on product spaces.

Connections between randomness, complexity, and Bayesian statistics.

Abstract

We study algorithmic randomness and monotone complexity on product of the set of infinite binary sequences. We explore the following problems: monotone complexity on product space, Lambalgen's theorem for correlated probability, classification of random sets by likelihood ratio tests, decomposition of complexity and independence, Bayesian statistics for individual random sequences. Formerly Lambalgen's theorem for correlated probability is shown under a uniform computability assumption in [H. Takahashi Inform. Comp. 2008]. In this paper we show the theorem without the assumption.