The Range of a Simple Random Walk on $\mathbb{Z}$: An Elementary Combinatorial Approach

Bernhard A Moser

doi:10.37236/4106

Abstract

Two different elementary approaches for deriving an explicit formula for the distribution of the range of a simple random walk on $\mathbb{Z}$ of length $n$ are presented. Both of them rely on Hermann Weyl's discrepancy norm, which equals the maximal partial sum of the elements of a sequence. By this the original combinatorial problem on $\mathbb{Z}$ can be turned into a known path-enumeration problem on a bounded lattice. The solution is provided by means of the adjacency matrix $\mathbf Q_d$ of the walk on a bounded lattice $(0,1,\ldots,d)$. The second approach is algebraic in nature, and starts with the adjacency matrix $\mathbf{Q_d}$. The powers of the adjacency matrix are expanded in terms of products of non-commutative left and right shift matrices. The representation of such products by means of the discrepancy norm reveals the solution directly.

Highlights

The problem of determining the distribution of the range of a simple random walk has been treated extensively in the literature
Two different elementary approaches for deriving an explicit formula for the distribution of the range of a simple random walk on Z of length n are presented. Both of them rely on Hermann Weyl’s discrepancy norm, which equals the maximal partial sum of the elements of a sequence. By this the original combinatorial problem on Z can be turned into a known path-enumeration problem on a bounded lattice
The solution is provided by means of the adjacency matrix Qd of the walk on a bounded lattice (0, 1, . . . , d)

Summary

Introduction

The problem of determining the distribution of the range of a simple random walk has been treated extensively in the literature. Feller [5] computes the distribution of the range of a standard Brownian motion and derives estimates for the discrete case. In this article he points out that the problem of finding exact formulae for the distribution of the range is difficult to solve in the discrete case. It is shown that a product of these non-commutative matrices can be represented in terms of the discrepancy norm of the sequence of the corresponding signs, −1 and +1, respectively. This leads to the intuitive Lost Walker Lemma, which immediately provides the solution

Discrepancy Norm

Geometric Approach

Algebraic Approach

Asymmetric Walks

Conclusion