  Given the increments of a simple symmetric random walk $(X_n)_{n\ge0}$, we
characterize all possible ways of recycling these increments into a simple
symmetric random walk $(Y_n)_{n\ge0}$ adapted to the filtration of
$(X_n)_{n\ge0}$. We study the long term behavior of a suitably normalized
two-dimensional process $((X_n,Y_n))_{n\ge0}$. In particular, we provide
necessary and sufficient conditions for the process to converge to a
two-dimensional Brownian motion (possibly degenerate). We also discuss cases in
which the limit is not Gaussian. Finally, we provide a simple necessary and
sufficient condition for the ergodicity of the recycling transformation, thus
generalizing results from Dubins and Smorodinsky (1992) and Fujita (2008), and
solving the discrete version of the open problem of the ergodicity of the
general L\'evy transformation (see Mansuy and Yor, 2006).
