A random walk analogue of Lévy’s Theorem

Abstract

In this paper we will give a simple symmetric random walk analogue of Lévy’s Theorem. We will give a new definition of a local time of the simple symmetric random walk. We apply a discrete Itô formula to some absolute value like function to obtain a discrete Tanaka formula. Results in this paper rely upon a discrete Skorokhod reflection argument. This random walk analogue of Lévy’s theorem was already obtained by G. Simons ([14]) but it is still worth noting because we will use a discrete stochastic analysis to obtain it and this method is applicable to other research. We note some connection with previous results by Csáki, Révész, Csörgő and Szabados. Finally we observe that the discrete Lévy transformation in the present version is not ergodic. Lastly we give a Lévy-type theorem for simple nonsymmetric random walk using a discrete bang-bang process.