Electronic Journal of Probability

Limit theorems for Parrondo's paradox

S Ethier and Jiyeon Lee

Full-text: Open access

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819523

Digital Object Identifier
doi:10.1214/EJP.v14-684

Mathematical Reviews number (MathSciNet)
MR2540850

Zentralblatt MATH identifier
1190.60060

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

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

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

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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References

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