Bootstrap random walks

Consider a one dimensional simple random walk X = (X_n)_n≥0. We form a new simple symmetric random walk Y = (Y_n)_n≥0 by taking sums of products of the increments of X and study the two-dimensional walk (X,Y) = ((X_n, Y_n))_n≥0. 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}).