Uniform test of algorithmic randomness over a general space
<|reference_start|>Uniform test of algorithmic randomness over a general space: 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).<|reference_end|>
@article{gacs2003uniform,
    title={Uniform test of algorithmic randomness over a general space},
    author={Peter Gacs},
    journal={Theoretical Computer Science 341 (2005) 91-137},
    year={2003},
    archivePrefix={arXiv},
    eprint={cs/0312039},
    primaryClass={cs.CC}
}