  The paper obtains analytical results for the asymptotic properties of Model
Selection Criteria -- widely used in practice -- for a general family of hidden
Markov models (HMMs), thereby substantially extending the related theory beyond
typical i.i.d.-like model structures and filling in an important gap in the
relevant literature. In particular, we look at the Bayesian and Akaike
Information Criteria (BIC and AIC) and the model evidence. In the setting of
nested classes of models, we prove that BIC and the evidence are strongly
consistent for HMMs (under regularity conditions), whereas AIC is not weakly
consistent. Numerical experiments support our theoretical results.
