  A quantum walker moves on the integers with four extra degrees of freedom,
performing a coin-shift operation to alter its internal state and position at
discrete units of time. The time evolution is described by a unitary process.
We focus on finding the limit probability law for the position of the walker
and study it by means of Fourier analysis. The quantum walker exhibits both
localization and a ballistic behavior. Our two results are given as limit
theorems for a 2-period time-dependent walk and they describe the location of
the walker after it has repeated the unitary process a large number of times.
The theorems give an analytical tool to study some of the Parrondo type
behavior in a quantum game which was studied by J. Rajendran and C. Benjamin by
means of very nice numerical simulations [1]. With our analytical tools at hand
we can easily explore the "phase space" of parameters of one of the games,
similar to the winning game in their papers. We include numerical evidence that
our two games, similar to theirs, exhibit a Parrondo type paradox.
