As well known, cumulant expansion is an alternative way to moment expansion to fully characterize probability distributions provided all the moments exist. If this is not the case, the so-called escort mean values (or q-moments) have been proposed to characterize probability densities with divergent moments [C. Tsallis et al., J. Math. Phys. 50, 043303 (2009)]. We introduce here a new mathematical object, namely, the q-cumulants, which, in analogy to the cumulants, provide an alternative characterization to that of the q-moments for the probability densities. To illustrate the technical details of the procedure, we apply this new scheme to further study a recently proposed family of scale-invariant discrete probabilistic models [A. Rodríguez et al., J. Stat. Mech.: Theory Exp. 2008, P09006; R. Hanel et al., Eur. Phys. J. B 72, 263 (2009)] having q-Gaussians as limiting probability distributions.