  In this paper, we study the full statistics of photons transmitted through an
optical cavity coupled to nanomechanical motion. We analyze the entire temporal
evolution of the photon correlations, the Fano factor, and the effects of
strong laser driving, all of which show pronounced features connected to the
mechanical backaction. In the regime of single-photon strong coupling, this
allows us to predict a transition from sub-Poissonian to super-Poissonian
statistics for larger observation time intervals. Furthermore, we predict
cascades of transmitted photons triggered by multi-photon transitions. In this
regime, we observe Fano factors that are drastically enhanced due to the
mechanical motion.
