In this paper we consider non parametric finite translation mixtures. We
prove that all the parameters of the model are identifiable as soon as the
matrix that defines the joint distribution of two consecutive latent variables
is non singular and the translation parameters are distinct. Under this
assumption, we provide a consistent estimator of the number of populations, of
the translation parameters and of the distribution of two consecutive latent
variables, which we prove to be asymptotically normally distributed under mild
dependency assumptions. We propose a non parametric estimator of the unknown
translated density. In case the latent variables form a Markov chain (Hidden
Markov models), we prove an oracle inequality leading to the fact that this
estimator is minimax adaptive over regularity classes of densities
