  Since the earliest implementations of the various GW approximations and
cumulant expansion in the calculations of quasiparticle propagators and
spectra, several attempts have been made to combine the advantageous properties
and results of these two theoretical approaches. While the GW plus cumulant
approach has proven successful in interpreting photoemission spectroscopy data
in solids, the formal connection between the two methods has not been
investigated in detail. By introducing a general bijective integral
representation of the cumulants, we can rigorously identify at which point
these two approximations can be connected for the paradigmatic model of
quasiparticle interaction with the dielectric response of the system that has
been extensively exploited in recent interpretations of the satellite
structures in photoelectron spectra. We establish a protocol for consistent
practical implementation of the thus established GW+cumulant scheme, and
illustrate it by comprehensive state-of-the-art first-principles calculations
of intrinsic angle-resolved photoemission spectra from Si valence bands.
