Persistence probabilities of autoregressive chains with continuous innovations

TL;DR

This paper analyzes the persistence probabilities of autoregressive chains with continuous innovations, revealing their geometric nature, factorization properties, and distributional characteristics of first passage times under various conditions.

Contribution

It generalizes Baxter-Spitzer factorization to autoregressive chains with continuous innovations and characterizes first passage time distributions for different drift and innovation types.

Findings

Persistence probabilities are compound-geometric for positive drifts.

First passage times are log-convex or log-concave depending on drift and innovation log-concavity.

Bi-exponential innovations lead to an additive exponential law factorization.

Abstract

We consider the persistence probabilities of an autoregressive chain of order one with continuous innovations. In the case of positive drifts, we show that these persistence probabilities are compound-geometric and satisfy a Baxter-Spitzer factorization generalizing that of the random walk. In the case of negative drifts, we exhibit a discrete Van Dantzig problem, which implies that the Baxter-Spitzer factorization never happens, except in a degenerate case. For positive drifts and log-concave innovations, we show that the first passage time in $(-\infty,0)$ has a log-convex distribution, whereas in the case of negative drifts and log-convex innovations on ${\mathbb R}^+$, it has a log-concave distribution. The case of the bi-exponential innovations is studied in detail, which leads for positive drifts to an additive factorization of the exponential law.