The maximum of a symmetric next neighbor walk on the nonnegative integers

Abstract

We consider a one-dimensional discrete symmetric random walk with a reflecting boundary at the origin. Generating functions are found for the two-dimensional probability distribution P{S_n = x, max_1≤j≤nS_n = 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) one-dimensional distribution for S_n = x, but more importantly that for max_1≤j≤nS_j = 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.