Bootstrap Random Walks

TL;DR

This paper introduces a novel method of constructing a two-dimensional walk from a simple random walk by summing products of its increments, demonstrating recurrence and convergence to independent Brownian motions, and extends these results to finite increment sets.

Contribution

It presents a new approach to creating multi-dimensional processes from simple random walks, establishing their recurrence, independence, and convergence properties, including finite set increments.

Findings

The constructed walk is recurrent.

It converges to a two-dimensional Brownian motion with independent components.

The results extend to finite set increments.

Abstract

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 $\{-1,+1\}$).