Bootstrap random walks

Consider a one dimensional simple random walk X=(Xn)n≥0X=(Xn)n≥0. We form a new simple symmetric random walk Y=(Yn)n≥0Y=(Yn)n≥0 by taking sums of products of the increments of XX and study the two-dimensional walk (X,Y)=((Xn,Yn))n≥0(X,Y)=((Xn,Yn))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}{−1,+1}).