  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.
