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We establish exact formulae for the persistence probabilities of an AR(1)sequence with symmetric uniform innovations in terms of certain families ofpolynomials, most notably a family introduced by Mallows and Riordan asenumerators of finite labeled trees when ordered by inversions. The connectionof these polynomials with the volumes of certain polytopes is also discussed.Two further results provide factorizations of general AR(1) models, one fornegative drifts with continuous innovations, and one for positive drifts withcontinuous and symmetric innovations. The second factorization extends aclassical universal formula of Sparre Andersen for symmetric random walks. Ourresults also lead to explicit asymptotic estimates for the persistenceprobabilities.

Persistence for a class of order-one autoregressive processes and Mallows-Riordan polynomials