Electronic Journal of Probability

Limit theorems for Parrondo's paradox

S Ethier and Jiyeon Lee

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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.

Article information

Electron. J. Probab., Volume 14 (2009), paper no. 62, 1827-1862.

Accepted: 2 September 2009
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60F05: Central limit and other weak theorems

Parrondo's paradox Markov chain strong law of large numbers central limit theorem strong mixing property fundamental matrix spectral representation

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Ethier, S; Lee, Jiyeon. Limit theorems for Parrondo's paradox. Electron. J. Probab. 14 (2009), paper no. 62, 1827--1862. doi:10.1214/EJP.v14-684. https://projecteuclid.org/euclid.ejp/1464819523

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