A generalization of the cumulant expansion. Application to a scale-invariant probabilistic model

Abstract

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.