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

Abstract

This work establishes exact formulae for the persistence probabilities pk(θ)=P[Y1⩾0,…,Yk⩾0] of an AR(1) sequence Yn=θYn−1+Xn, n=1,2,… with parameter θ∈R﹨(12,2) and symmetric uniform innovations Xn. The formulae are in terms of certain polynomials, most notably a family that arises in the case −1<θ<12 and was introduced by Mallows and Riordan in the very different context of counting finite labeled trees when ordered by inversions. The connection of these polynomials with the volumes of certain polytopes is also discussed. Two further results establish convolution-type factorizations in terms of the pk(θ) and their involutive conjugates pk(1/θ) for k=1,…,n and n⩾1. Regarding exact formulae for the pn(θ), these results are used for the cases θ<−1 and θ>2, but they are actually derived under more general conditions and therefore of independent interest, namely, one for AR(1) models with negative θ and continuous innovations, and a second one for AR(1) models with positive θ and continuous and symmetric innovations, the latter extending a classical universal formula of Sparre Andersen for symmetric random walks. We further explain why the case 12<θ<2 does not allow exact formulae for the pn(θ) as in the other cases and show that our results also lead to explicit asymptotic estimates for these probabilities.