The maximum of a symmetric next neighbor walk on the non-negative integers

Abstract

We consider a one-dimensional discrete symmetric random walk with a reflecting boundary at the origin. Generating functions are found for the 2- dimensional probability distribution P{Sn = x,max1?j?n Sn = a} of being at position x after n steps, while the maximal location that the walker has achieved during these n steps is a. We also obtain the familiar (marginal) 1-dimensional distribution for Sn = x, but more importantly that for max1?j?n Sj = a asymptotically at fixed a2/n. We are able to compute and compare the expectations and variances of the two one-dimensional distributions, finding that they have qualitatively similar forms, but differ quantitatively in the anticipated fashion.