Mathematics Colloquia and Seminars

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From a simple random walk one may obtain a random permutation of indices [1,n] via the lexicographic ordering first on the value of the walk at a given time, and second by the time itself. We demonstrate that by rearranging the increments of a random walk bridge according to this quantile permutation, we obtain a Dyck path. Passing to a Brownian limit gives a novel proof and a generalization of a theorem of Jeulin (1985) concerning Brownian local times.