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Stationary ergodic processes with finite alphabets are estimated by finitememory processes from a sample, an n-length realization of the process, wherethe memory depth of the estimator process is also estimated from the sampleusing penalized maximum likelihood (PML). Under some assumptions on thecontinuity rate and the assumption of non-nullness, a rate of convergence in$\bar{d}$-distance is obtained, with explicit constants. The result requires ananalysis of the divergence of PML Markov order estimators for not necessarilyfinite memory processes. This divergence problem is investigated in moregenerality for three information criteria: the Bayesian information criterionwith generalized penalty term yielding the PML, and the normalized maximumlikelihood and the Krichevsky-Trofimov code lengths. Lower and upper bounds onthe estimated order are obtained. The notion of consistent Markov orderestimation is generalized for infinite memory processes using the concept oforacle order estimates, and generalized consistency of the PML Markov orderestimator is presented.

Divergence rates of Markov order estimators and their application to statistical estimation of stationary ergodic processes