Limit theorems for Parrondo's paradox

Details Limit theorems for Parrondo's paradox Limit theorems for Parrondo's paradox

2009-02-13

arxiv.org/abs/0902.2368v2

Electronic Journal of Probability 14 (2009), 1827-1862

Mathematics

Probability

39 pages, 1 figure. Version 2 uses term "capital-dependent" instead of "profit-dependent," discusses an alternative approach t

Scientific paper

That there exist two losing games that can be combined, either by random mixture or by nonrandom alternation, to form a winning game is known as Parrondo's paradox. We establish a strong law of large numbers and a central limit theorem for the Parrondo player's sequence of profits, both in a one-parameter family of capital-dependent games and in a two-parameter family of history-dependent games, with the potentially winning game being either a random mixture or a nonrandom pattern of the two losing games. We derive formulas for the mean and variance parameters of the central limit theorem in nearly all such scenarios; formulas for the mean permit an analysis of when the Parrondo effect is present.

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