Limit theorems and ergodicity for general bootstrap random walks

TL;DR

This paper characterizes all ways to recycle increments of a simple symmetric random walk into another such walk, studies their long-term behavior, and provides conditions for convergence to Brownian motion and ergodicity.

Contribution

It offers a complete characterization of recycling transformations, analyzes their asymptotic behavior, and solves the open problem of ergodicity for the general Lévy transformation.

Findings

Conditions for convergence to two-dimensional Brownian motion

Cases where the limit is non-Gaussian

Necessary and sufficient conditions for ergodicity

Abstract

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).