Open Access
2009 Limit theorems for Parrondo's paradox
S Ethier, Jiyeon Lee
Author Affiliations +
Electron. J. Probab. 14: 1827-1862 (2009). DOI: 10.1214/EJP.v14-684


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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S Ethier. Jiyeon Lee. "Limit theorems for Parrondo's paradox." Electron. J. Probab. 14 1827 - 1862, 2009.


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

zbMATH: 1190.60060
MathSciNet: MR2540850
Digital Object Identifier: 10.1214/EJP.v14-684

Primary: 60J10
Secondary: 60F05

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

Vol.14 • 2009
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