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Limit theorems for Parrondo’s paradox. (English) Zbl 1190.60060

Summary: 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.

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60F05 Central limit and other weak theorems
60F15 Strong limit theorems
91A60 Probabilistic games; gambling
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