Given an autoregressive process X of order p
(i.e.
Xn = a1Xn−1 + ··· + apXn−p + Yn
where the random variables Y1, Y2,... 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 a1,...,
ap and the distribution of
Y1, 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.