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

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 are in terms certain polynomials, most notably a family that arises case −1<θ<12 was introduced by Mallows Riordan very different context counting finite labeled trees when ordered inversions. connection these polynomials volumes polytopes is also discussed. Two further results establish convolution-type factorizations pk(θ) their involutive conjugates pk(1/θ) k=1,…,n n⩾1. Regarding pn(θ), used cases θ<−1 θ>2, but they actually derived under more general conditions therefore independent interest, namely, one models negative θ continuous innovations, second positive latter extending classical universal formula Sparre Andersen random walks. We explain why 12<θ<2 does not allow pn(θ) as other show our lead to explicit asymptotic estimates probabilities.