The different asymptotic regimes of nearly unstable autoregressive processes

Abstract

We extend the results of [14, 27, 29] about the convergence of nearly unstable AR(p) processes to the infinite order case. To do so, we proceed as in [19, 20] by using limit theorems for some well chosen geometric sums. We prove that when the coefficients sequence has a light tail, nearly unstable AR(∞) processes behave as Ornstein-Uhlenbeck models. However, in the heavy tail case, we show that fractional diffusions arise as limiting laws for such processes.