The Different Asymptotic Regimes of Nearly Unstable Autoregressive Processes.

Summary 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(\(\infty \)) processes behave as Ornstein-Uhlenbeck models. However, in the heavy tail case, we show that fractional diffusions arise as limiting laws for such processes.