A conditional limit theorem is derived for a certain class of stochastic processes whose distributions constitute a nonergodic family. The limit theorem allows us to study the asymptotic behaviour under the conditional model of some standard statistical procedures by making use of results for ergodic families. Explosive Gaussian autoregressive processes are studied in some detail. Here the conditional process is shown to be a nonexplosive Gaussian autoregression bearing a simple relation to the original process. Some optimality results under the conditional model are given for estimators and tests based on the unconditional likelihood.
"Asymptotic Conditional Inference for Regular Nonergodic Models with an Application to Autoregressive Processes." Ann. Statist. 12 (1) 161 - 171, March, 1984. https://doi.org/10.1214/aos/1176346399