Persistence of one-dimensional AR(1)-processes

For a class of one-dimensional autoregressive sequences (Xn), we consider the tail behaviour of the stopping time T0=min{n≥1:Xn≤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 T0 and on the analytical Fredholm alternative. Using this method, we show that Px(T0=n)∼V(x)Rn0 for some 0<R0<1 and a positive R0-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.