  Consider a one dimensional simple random walk $X=(X_n)_{n\geq0}$. We form a
new simple symmetric random walk $Y=(Y_n)_{n\geq0}$ by taking sums of products
of the increments of $X$ and study the two-dimensional walk
$(X,Y)=((X_n,Y_n))_{n\geq0}$. We show that it is recurrent and when suitably
normalised converges to a two-dimensional Brownian motion with independent
components; this independence occurs despite the functional dependence between
the pre-limit processes. The process of recycling increments in this way is
repeated and a multi-dimensional analog of this limit theorem together with a
transience result are obtained. The construction and results are extended to
include the case where the increments take values in a finite set (not
necessarily $\{-1,+1\}$).
