  This paper deals with the problem of estimating the coupling constant
$\theta$ of a mixing quantum Markov chain. For a repeated measurement on the
chain's output we show that the outcomes' time average has an asymptotically
normal (Gaussian) distribution, and we give the explicit expressions of its
mean and variance. In particular we obtain a simple estimator of $\theta$ whose
classical Fisher information can be optimized over different choices of
measured observables. We then show that the quantum state of the output
together with the system, is itself asymptotically Gaussian and compute its
quantum Fisher information which sets an absolute bound to the estimation
error. The classical and quantum Fisher informations are compared in a simple
example. In the vicinity of $\theta=0$ we find that the quantum Fisher
information has a quadratic rather than linear scaling in output size, and
asymptotically the Fisher information is localised in the system, while the
output is independent of the parameter.
