Ethier, S. N.; Lee, Jiyeon Limit theorems for Parrondo’s paradox. (English) Zbl 1190.60060 Electron. J. Probab. 14, 1827-1862 (2009). Summary: 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. Cited in 7 Documents MSC: 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60F05 Central limit and other weak theorems 60F15 Strong limit theorems 91A60 Probabilistic games; gambling Keywords:Parrondo’s paradox; Markov chain; strong law of large numbers; central limit theorem; strong mixing property; fundamental matrix; spectral representation PDFBibTeX XMLCite \textit{S. N. Ethier} and \textit{J. Lee}, Electron. J. Probab. 14, 1827--1862 (2009; Zbl 1190.60060) Full Text: DOI arXiv EuDML EMIS