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A hidden Markov model with trends is a hidden Markov model whose emissiondistributions are translated by a trend that depends on the current hiddenstate and on the current time. Contrary to standard hidden Markov models, suchprocesses are not homogeneous and cannot be made homogeneous by a simplede-trending step. We show that when the trends are polynomial, the maximumlikelihood estimator is able to recover the trends together with the otherparameters and is strongly consistent. More precisely, the supremum norm of thedifference between the true trends and the estimated ones tends to zero.Numerical properties of the maximum likelihood estimator are assessed by asimulation study.

Consistency of the maximum likelihood estimator in hidden Markov models with trends