Charge transfer statistics of a molecular quantum dot with a vibrational degree of freedom

We analyze the full counting statistics of a single-site quantum dot coupled to a local Holstein phonon for arbitrary transmission and weak electron-phonon coupling. We identify explicitly the contributions due to quasielastic and inelastic transport processes in the cumulant generating function and discuss their influence on the transport properties of the dot. We find that in the low-energy sector, i.e., for bias voltage and phonon frequency much smaller than the dot-electrode contact transparency, the inelastic term causes a sign change in the shot noise correction at certain universal values of the transmission. Furthermore, we show that when the correction to the current due to inelastic processes vanishes, all the odd order cumulants vanish as well.