Mixing convergence of LSE for supercritical AR(2) processes with Gaussian innovations using random scaling

TL;DR

This paper proves the mixing convergence of the least squares estimator for supercritical AR(2) processes with Gaussian innovations, using random scaling to obtain a limit distribution concentrated on a specific ray.

Contribution

It introduces a novel random scaling approach to establish the asymptotic distribution of the LSE in supercritical AR(2) models with Gaussian noise.

Findings

LSE converges in distribution to a normal distribution on a ray

Random scaling effectively captures the asymptotic behavior

Results apply to processes with roots of different absolute values

Abstract

We prove mixing convergence of the least squares estimator of autoregressive parameters for supercritical autoregressive processes of order 2 with Gaussian innovations having real characteristic roots with different absolute values. We use an appropriate random scaling such that the limit distribution is a two-dimensional normal distribution concentrated on a one-dimensional ray determined by the characteristic root having the larger absolute value.