The consistency of the BIC Markov order estimator

Abstract

We show that the BIC (Bayesian information criterion) estimator of the order of a Markov chain (with finite alphabet) gives the correct order, eventually almost surely as the sample size goes to /spl infin/, thereby strengthening earlier consistency results that assumed an apriori bound on the order. A key tool is a strong typicality result for Markov sample paths. We also show that the Bayesian or MDL estimator, of which the BIC estimator is regarded as an approximation, fails to be consistent for the uniformly distributed i.i.d. process.