Uniform test of algorithmic randomness over a general space

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-Löf) 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. The issues are the following:• 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. • 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).

Applications of Effective Probability Theory to Martin-Löf Randomness

“…Its generalization to abstract spaces has been investigated in [ZL70,HW03,Gác05,HR09b]. We follow the approaches [Gác05,HR09b] developed on any computable probability space (X, µ).…”

“…This function is known to be the logarithm of a universal integrable µ-test, which means that for every integrable µ-test t there is a constant a such that log t ≤ a + d µ . On the other hand, every computable probability space admits a universal integrable test t µ (see [Gác05,HR09b]). Generalizing Davie, let us define K c := {x : t µ (x) ≤ 2 c }.…”

“…In [HR09a], working in the context of computable probability spaces (to which Martin-Löf randomness has been recently extended, see [Gác05,HR09b]), effective versions of measure-theoretic notions were examined and another contribution of algorithmic randomness to probability theory was developed: the setting of a new framework for computability adapted to the probabilistic context. This was achieved by making a fundamental use of the existence of a universal Martin-Löf test to endow the space with what we call the Martin-Löf layering.…”

Cone avoidance and randomness preservation

“…Our approach is derived from the one in [7,26] augmented by ideas from [23, §3.3]. A more elaborate and general coding-free approach may be found in [11].…”

How Much Randomness Is Needed for Statistics?

“…Reimann and Slaman [14,arXiv:0802.2705, Definition 3.2.] defined a real x to be µ-random if, for some oracle z computing µ, the real x is µ-random relative to z. Levin [9] and Gács [6] use a uniform test, which is a left-c.e. function u :…”