We show that large-scale typicality of Markov sample paths implies that the likelihood ratio statistic satisfies a law of iterated logarithm uniformly to the same scale. As a consequence, the penalized likelihood Markov order estimator is strongly consistent for penalties growing as slowly as log log n when an upper bound is imposed on the order which may grow as rapidly as log n. Our method of proof, using techniques from empirical process theory, does not rely on the explicit expression for the maximum likelihood estimator in the Markov case and could therefore be applicable in other settings.
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van Handel, R. On the minimal penalty for Markov order estimation. Probab. Theory Relat. Fields 150, 709–738 (2011). https://doi.org/10.1007/s00440-010-0290-y
- Order estimation
- Uniform law of iterated logarithm
- Martingale inequalities
- Empirical process theory
- Large-scale typicality
- Markov chains
Mathematics Subject Classification (2000)
- Primary 62M05
- Secondary 60E15