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

TL;DR

This paper analyzes a symmetric random walk with a reflecting boundary, deriving generating functions for the joint distribution of position and maximum, and providing asymptotic behaviors and moments of these distributions.

Contribution

It introduces explicit generating functions for the joint distribution of position and maximum in a symmetric walk with boundary, and explores their asymptotic properties.

Findings

Derived generating functions for joint distributions.

Obtained asymptotic distributions for maximum.

Compared expectations and variances of position and maximum.

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.