Persistence of one-dimensional AR(1)-sequences

TL;DR

This paper studies the tail behavior of the stopping time for one-dimensional AR(1) processes, introducing a new analytical method based on renewal decomposition and Fredholm theory to characterize the decay rate.

Contribution

It proposes a novel approach using renewal-type decomposition and Fredholm alternative to analyze the tail behavior of AR(1) processes, providing sharp conditions for exponential decay.

Findings

The tail probability decays exponentially with rate R_0.

The method characterizes the asymptotic form of the tail probability.

Fatter innovation tails lead to non-exponential decay factors.

Abstract

For a class of one-dimensional autoregressive processes $(X_n)$ we consider the tail behaviour of the stopping time $T_0=\min \lbrace n\geq 1: X_n\leq 0 \rbrace$. We discuss existing general analytical approaches to this and related problems and propose a new one, which is based on a renewal-type decomposition for the moment generating function of $T_0$ and on the analytical Fredholm alternative. Using this method, we show that $\mathbb{P}_x(T_0=n)\sim V(x)R_0^n$ for some $0<R_0<1$ and a positive $R^{-1}_0$-harmonic function $V$. Further we prove that our conditions on the tail behaviour of the innovations are sharp in the sense that fatter tails produce non-exponential decay factors.