Persistence for a class of order-one autoregressive processes and Mallows-Riordan polynomials - Archive ouverte HAL Access content directly
Journal Articles Advances in Applied Mathematics Year : 2022

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

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Abstract

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.
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Dates and versions

hal-03469594 , version 1 (07-12-2021)

Identifiers

  • HAL Id : hal-03469594 , version 1

Cite

Gerold Alsmeyer, Alin Bostan, Kilian Raschel, Thomas Simon. Persistence for a class of order-one autoregressive processes and Mallows-Riordan polynomials. Advances in Applied Mathematics, 2022. ⟨hal-03469594⟩
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