Uniform test of algorithmic randomness over a general space

TL;DR

This paper extends the algorithmic randomness theory to general spaces, including non-compact ones, by defining a universal uniform test and exploring its properties and implications.

Contribution

It introduces a framework for defining and analyzing randomness tests in arbitrary spaces, generalizing previous results to non-compact and continuous spaces.

Findings

Existence of universal tests in general spaces

Conservation of randomness holds in the new framework

The deficiency of randomness generalizes complexity in continuous spaces

Abstract

The algorithmic theory of randomness is well developed when the underlying space is the set of finite or infinite sequences and the underlying probability distribution is the uniform distribution or a computable distribution. These restrictions seem artificial. Some progress has been made to extend the theory to arbitrary Bernoulli distributions (by Martin-Loef), and to arbitrary distributions (by Levin). We recall the main ideas and problems of Levin's theory, and report further progress in the same framework. - We allow non-compact spaces (like the space of continuous functions, underlying the Brownian motion). - The uniform test (deficiency of randomness) d_P(x) (depending both on the outcome x and the measure P should be defined in a general and natural way. - We see which of the old results survive: existence of universal tests, conservation of randomness, expression of tests in terms of description complexity, existence of a universal measure, expression of mutual information as "deficiency of independence. - The negative of the new randomness test is shown to be a generalization of complexity in continuous spaces; we show that the addition theorem survives. The paper's main contribution is introducing an appropriate framework for studying these questions and related ones (like statistics for a general family of distributions).