  We consider parameter estimation in finite hidden state space Markov models
with time-dependent inhomogeneous noise, where the inhomogeneity vanishes
sufficiently fast. Based on the concept of asymptotic mean stationary processes
we prove that the maximum likelihood and a quasi-maximum likelihood estimator
(QMLE) are strongly consistent. The computation of the QMLE ignores the
inhomogeneity, hence, is much simpler and robust. The theory is motivated by an
example from biophysics and applied to a Poisson- and linear Gaussian model.
