The modified, discrete, Levy-transformation is Bernoulli

Abstract

From the absolute value of a martingale, X, there is a unique increasing process that can be subtracted so as to obtain a martingale, Y. Paul Levy discovered that if X is Brownian motion, B, then Y, too, is a Brownian motion. Equivalently, Levy found that the transformation that maps B to Y is measure-preserving. Whether it is ergodic, a question raised by Marc Yor, is open. Here, the natural analogue of Levy's transformation for the symmetric random walk is modified and, thus modified, is shown to be measure-preserving. The ergodicity of this transformation is then established by showing that it is isomorphic to the one-sided, Bernoulli shift-transformation associated with a sequence of independent random variables, each uniformly distributed on the unit interval.