Persistence of one-dimensional AR(1)-sequences

Abstract

For a class of one-dimensional autoregressive processes $(X_n)$ we consider the tail behaviour of the stopping time $T_0=\min \{n\ge 1:X_n\le 0\}$. 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 $R_0^{-1}$-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.

30 pages