The different asymptotic regimes of nearly unstable autoregressive processes

We extend classical results about the convergence of nearly unstable AR(p) processes to the infinite order case. To do so, we proceed as in recent works about Hawkes processes by using limit theorems for some well chosen geometric sums. We prove that when the coefficients sequence has a light tail, infinite order nearly unstable autoregressive processes behave as Ornstein-Uhlenbeck models. However, in the heavy tail case, we show that fractional diffusions arise as limiting laws for such processes.