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The Range of a Simple Random Walk on Z: An Elementary Combinatorial Approach

unpublished

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 Q d of the walk on a bounded lattice

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