  We study inelastic effects on the electronic current noise in molecular
junctions, due to the coupling between transport electrons and vibrational
degrees of freedom. Using a full counting statistics approach based on the
generalized Keldysh Green's function technique, we calculate in an unified
manner both the mean current and the zero-frequency current noise. For
multilevel junctions with weak electron-phonon coupling, we give analytical
formulas for the lowest order inelastic corrections to the noise in terms of
universal temperature- and voltage-dependent functions and junction-dependent
prefactors, which can be evaluated microscopically, e.g. with ab-initio
methodologies. We identify distinct terms corresponding to the mean-field
contribution to noise and to the vertex corrections, and we show that the
latter contribute substantially to the inelastic noise. Finally, we illustrate
our results by a simple model of two electronic levels which are mutually
coupled by the electron-phonon interaction and show that the inelastic noise
spectroscopy is a sensitive diagnostic tool.
