Two new Markov order estimators
  We present two new methods for estimating the order (memory depth) of a finite alphabet Markov chain from observation of a sample path. One method is based on entropy estimation via recurrence times of patterns, and the other relies on a comparison of empirical conditional probabilities. The key to both methods is a qualitative change that occurs when a parameter (a candidate for the order) passes the true order. We also present extensions to order estimation for Markov random fields. 
Yuval Peres and Paul Shields