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

TL;DR

This paper introduces q-cumulants as a new mathematical tool to characterize probability densities, especially useful for scale-invariant models with divergent moments, extending the cumulant concept to non-traditional distributions.

Contribution

The paper proposes q-cumulants as a novel extension of cumulants, applicable to probability densities with divergent moments, and demonstrates their use on scale-invariant models with q-Gaussian limits.

Findings

Q-cumulants provide an alternative to q-moments for characterizing distributions.

Application to scale-invariant models with q-Gaussian limits.

Enhances understanding of probability distributions with divergent moments.

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. 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.