  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.
