@article{EJP684,
author = {S Ethier and Jiyeon Lee},
title = {Limit theorems for Parrondo's paradox},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {14},
year = {2009},
keywords = {Parrondo's paradox, Markov chain, strong law of large numbers, central limit theorem, strong mixing property, fundamental matrix, spectral representation.},
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.},
pages = {no. 62, 1827-1862},
issn = {1083-6489},
doi = {10.1214/EJP.v14-684},
url = {http://ejp.ejpecp.org/article/view/684}}