Survival probabilities of autoregressive processes

Given an autoregressive process X of order p (i.e. X_n = a_1 X_{n-1} + …+ a_p X_{n_p} + Y_n where the random variables Y_1, Y_2, … are i.i.d.), we study the asymptotic behaviour of the probability that the process does not exceed a constant barrier up to time N (survival or persistence probability). Depending on the coefficients a_1,…,a_p and the distribution of Y_1, we state conditions under which the survival probability decays polynomially, faster than polynomially or converges to a positive constant. Special emphasis is put on AR(2) processes.