  We consider the problem of estimating an arbitrary dynamical parameter of an
quantum open system in the input-output formalism. For irreducible Markov
processes, we show that in the limit of large times the system-output state can
be approximated by a quantum Gaussian state whose mean is proportional to the
unknown parameter. This approximation holds locally in a neighbourhood of size
$t^{-1/2}$ in the parameter space, and provides an explicit expression of the
asymptotic quantum Fisher information in terms of the Markov generator.
  Furthermore we show that additive statistics of the counting and homodyne
measurements also satisfy local asymptotic normality and we compute the
corresponding classical Fisher informations. The mathematical theorems are
illustrated with the examples of a two-level system and the atom maser.
  Our results contribute towards a better understanding of the statistical and
probabilistic properties of the output process, with relevance for quantum
control engineering, and the theory of non-equilibrium quantum open systems.
