  We study the full-counting statistics of charges transmitted through a
single-level quantum dot weakly coupled to a local Einstein phonon which causes
fluctuations in the dot energy. An analytic expression for the cumulant
generating function, accurate up to second order in the electron-phonon
coupling and valid for finite voltages and temperatures, is obtained in the
extended wide-band limit. The result accounts for nonequilibrium phonon
distributions induced by the source-drain bias voltage, and concomitantly
satisfies the fluctuation theorem. Extending the counting field to the complex
plane, we investigate the locations of possible singularities of the cumulant
generating function, and exploit them to identify regimes in which the electron
transfer is affected differently by the coupling to the phonons. Within a
large-deviation analysis, we find a kink in the probability distribution,
analogous to a first-order phase transition in thermodynamics, which would be a
unique hallmark of the electron-phonon correlations. This kink reflects the
fact that although inelastic scattering by the phonons once the voltage exceeds
their frequency can scatter electrons opposite to the bias, this will never
generate current flowing against the bias at zero temperature, in accordance
with the fluctuation theorem.
