Elementary Proof of a Theorem of Jean Ville

Elliott H. Lieb; Daniel Osherson; Scott Weinstein

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Elementary Proof of a Theorem of Jean Ville

Elliott H. Lieb·Daniel Osherson·Scott Weinstein

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Abstract

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.

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Elliott H. Lieb, Daniel Osherson, Scott Weinstein. 2006-07-11. Elementary Proof of a Theorem of Jean Ville. https://arxiv.org/abs/cs/0607054

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