The different asymptotic regimes of nearly unstable autoregressive processes

TL;DR

This paper investigates the asymptotic behavior of nearly unstable autoregressive processes of infinite order, revealing their convergence to Ornstein-Uhlenbeck models or fractional diffusions depending on the tail properties of their coefficients.

Contribution

It extends classical convergence results to infinite order AR processes, distinguishing between light and heavy tail coefficient sequences and identifying their limiting processes.

Findings

Light tail coefficients lead to Ornstein-Uhlenbeck limits.

Heavy tail coefficients result in fractional diffusion limits.

Provides a unified framework for different asymptotic regimes.

Abstract

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.