Random Walks at Random Times

Dr. Paul Jung

Department of Mathematics
University of Alabama at Birmingham

6 April 2012
218 Shelby Center
3:00 (Refreshements at 2:30)

Abstract

A random walk in random scenery (RWRS) is a collective reward process where a random walker collects a random reward (or scenery) at each site it visits. If the walker visits a site multiple times, it collects the same reward many times thus leading to correlations in the collective reward process. Cohen and Samorodnitsky (2006) studied a certain renormalization of RWRS and proposed self-similar, symmetric α-stable processes, which generalize fractional Brownian motion as their scaling limits. The limiting processes have self-similarity exponents H>1/α.

We consider a modification in which a sign associated to the reward (scenery) alternates upon successive visits. The resulting process is what we call a random walk at a random time, and it generalizes the so-called iterated random walk. We will discuss weak convergence of the discrete processes to their scaling limits, and in particular, show that the alternating scenery leads to limiting processes which have self-similarity exponents H<1/α.