  Quantum trajectories are Markov processes modeling the evolution of a quantum
system subjected to repeated independent measurements. Under purification and
irreducibility assumptions, these Markov processes admit a unique invariant
measure - see Benoist et al. Probab. Theory Relat. Fields 2019. In this article
we prove, finer limit theorems such as Law of Large Numbers (LLN), Functional
Central Limit Theorem, Law of Iterated Logarithm and Moderate Deviation
Principle. The proof of the LLN is based on Birkhoff's ergodic theorem and an
analysis of harmonic functions. The other theorems are proved using martingale
approximation of empirical sums.
