On the Cs\'aki-Vincze transformation

TL;DR

This paper investigates the ergodic and asymptotic properties of the Csáki-Vincze transformation on simple random walks, proving its exactness and showing convergence to the Lévy transform of Brownian motion in a scaling limit.

Contribution

It establishes the exactness of the Csáki-Vincze transformation and describes its asymptotic behavior, including convergence to the Lévy transform in a continuous limit.

Findings

Proves the transformation is exact for simple random walks.

Provides a description of the lost information at each iteration.

Shows convergence of the transformation's iterations to the Lévy transform of Brownian motion.

Abstract

Cs aki and Vincze have de fined in 1961 a discrete transformation T which applies to simple random walks and is measure preserving. In this paper, we are interested in ergodic and assymptotic properties of T . We prove that T is exact : \cap_{k\geq 1} \sigma(T^k(S)) is trivial for each simple random walk S and give a precise description of the lost information at each step k. We then show that, in a suitable scaling limit, all iterations of T "converge" to the corresponding iterations of the continous L evy transform of Brownian motion. Some consequences are also derived from these two results.