The different asymptotic regimes of nearly unstable autoregressive processes

2015/02/23 by Thibault Jaisson, Jaisson, Thibault, Mathieu Rosenbaum +1
Economics, Econometrics and Finance · Mathematics · #60F99 #FOS: Mathematics #Financial Risk and Volatility Modeling #Point processes and geometric inequalities #Statistics Theory (math.ST) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1502.06338

openalex publication_date 2015/02/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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