Motivated by previous work of Aurzada, Mukherjee and Zeitouni on persistence exponents of Markov chains and in particular autoregressive processes we consider the tail behaviour of the stopping time \(T_0 = min\{n \ge 1 : X_n \le 0\}\) for a class of one-dimensional autoregressive processes \((X_n)\). We discuss existing general analytic/probabilistic 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 analytic Fredholm alternative. Using this method, we show that \(P_x(T_0 = n) \sim V (x)R_0^n\) for some \(0 < R_0 < 1\). Furthermore we are able to prove convergence towards quasistationarity in our situation. (Joint work with G. Hinrichs and V. Wachtel)