  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. We illustrate this new scheme on a recently proposed
family of scale-invariant discrete probabilistic models [A. Rodriguez et al, J.
Stat. Mech. (2008) P09006; R. Hanel et al, Eur. Phys. J. B 72, 263268 (2009)]
having q-Gaussians as limiting probability distributions.
