  We study analytically the full counting statistics of charge transport
through single molecules, strongly coupled to a weakly damped vibrational mode.
The specifics of transport in this regime - a hierarchical sequence of
avalanches of transferred charges, interrupted by "quiet" periods - make the
counting statistics strongly non-Gaussian. We support our findings for the
counting statistics as well as for the frequency-dependent noise power by
numerical simulations, finding excellent agreement.
