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

We establish exact formulae for the persistence probabilities of an AR(1) sequence with symmetric uniform innovations in terms of certain families of polynomials, most notably a family introduced by Mallows and Riordan as enumerators of finite labeled trees when ordered by inversions. The connection of these polynomials with the volumes of certain polytopes is also discussed. Two further results provide general factorizations of AR(1) models with continuous symmetric innovations, one for negative and one for positive drift. The second factorization extends a classical universal formula of Sparre Andersen for symmetric random walks. Our results also lead to precise asymptotic estimates for the persistence probabilities. AMS 2020 subject classifications Primary 05C31; 60J05; Secondary 11B37; 30C15; 60F99 Gerold Alsmeyer Institute of Mathematical Stochastics, Department of Mathematics and Computer Science, University of Münster, Orléans-Ring 10, 48149 Münster, Germany. e-mail: gerolda@math.uni-muenster.de Alin Bostan Inria, Université Paris-Saclay, 1 rue Honoré d’Estienne d’Orves, 91120 Palaiseau, France. e-mail: alin.bostan@inria.fr Kilian Raschel CNRS, Laboratoire Angevin de Recherche en Mathématiques, Université d’Angers, 2 boulevard Lavoisier, 49045 Angers, France. e-mail: raschel@math.cnrs.fr Thomas Simon Laboratoire Paul Painlevé, Université de Lille, Cité Scientifique, 59655 Villeneuve d’Ascq, France. e-mail: thomas.simon@univ-lille.fr This project was partially funded by the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy EXC 2044–390685587 (Gerold Alsmeyer) and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under the Grant Agreement No 759702 (Kilian Raschel). Alin Bostan and Kilian Raschel were also supported in part by DeRerumNatura ANR-19-CE40-0018.

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