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We introduce a new method for proving central limit theorems for random walk on nilpotent groups. The method is illustrated in a local central limit theorem on the Heisenberg group, weakening the necessary conditions on the driving measure. As a second illustration, the method is used to study walks on the nn uni-upper triangular group with entries taken modulo p. The method allows sharp answers to the behavior of individual coordinates: coordinates immediately above the diagonal require order p^2 steps for randomness, coordinates on the second diagonal require order p steps; coordinates on the kth diagonal require order p^{2/k} steps.
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