Limit theorems for Parrondo's paradox
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.
Cite
@article{arxiv.0902.2368,
title = {Limit theorems for Parrondo's paradox},
author = {S. N. Ethier and Jiyeon Lee},
journal= {arXiv preprint arXiv:0902.2368},
year = {2009}
}
Comments
39 pages, 1 figure. Version 2 uses term "capital-dependent" instead of "profit-dependent," discusses an alternative approach to Section 5 suggested by a referee, and makes other minor changes