On large deviations in the Gaussian autoregressive process: stable, unstable and explosive cases

For the Gaussian autoregressive process, the asymptotic behaviour of the Yule-Walker estimator is totally different in the stable, unstable and explosive cases. We show that, irrespective of this trichotomy, this estimator shares quite similar large deviation properties in the three situations. However, in the explosive case, we obtain an unusual rate function with a discontinuity point at its minimum.