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

  • Bernhard A. Moser
Keywords: random walk, discrepancy norm, lattice path enumeration


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.

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