Elementary Proof of a Theorem of Jean Ville
<|reference_start|>Elementary Proof of a Theorem of Jean Ville: Considerable thought has been devoted to an adequate definition of the class of infinite, random binary sequences (the sort of sequence that almost certainly arises from flipping a fair coin indefinitely). The first mathematical exploration of this problem was due to R. Von Mises, and based on his concept of a "selection function." A decisive objection to Von Mises' idea was formulated in a theorem offered by Jean Ville in 1939. It shows that some sequences admitted by Von Mises as "random" in fact manifest a certain kind of systematicity. Ville's proof is challenging, and an alternative approach has appeared only in condensed form. We attempt to provide an expanded version of the latter, alternative argument.<|reference_end|>
@article{lieb2006elementary,
    title={Elementary Proof of a Theorem of Jean Ville},
    author={Elliott H. Lieb, Daniel Osherson and Scott Weinstein},
    journal={arXiv preprint arXiv:cs/0607054},
    year={2006},
    archivePrefix={arXiv},
    eprint={cs/0607054},
    primaryClass={cs.CC}
}