  This paper considers hidden Markov models where the observations are given as
the sum of a latent state which lies in a general state space and some
independent noise with unknown distribution. It is shown that these fully
nonparametric translation models are identifiable with respect to both the
distribution of the latent variables and the distribution of the noise, under
mostly a light tail assumption on the latent variables. Two nonparametric
estimation methods are proposed and we prove that the corresponding estimators
are consistent for the weak convergence topology. These results are illustrated
with numerical experiments.
