Comment on "Universal relation between skewness and kurtosis in complex dynamics".

In a recent paper [M. Cristelli, A. Zaccaria, and L. Pietronero, Phys. Rev. E 85, 066108 (2012)], the authors analyzed the relation between skewness and kurtosis for complex dynamical systems, and they identified two power-law regimes of non-Gaussianity, one of which scales with an exponent of 2 and the other with 4/3. They concluded that the observed relation is a universal fact in complex dynamical systems. In this Comment, we test the proposed universal relation between skewness and kurtosis with a large number of synthetic data, and we show that in fact it is not a universal relation and originates only due to the small number of data points in the datasets considered. The proposed relation is tested using a family of non-Gaussian distribution known as q-Gaussians. We show that this relation disappears for sufficiently large datasets provided that the fourth moment of the distribution is finite. We find that kurtosis saturates to a single value, which is of course different from the Gaussian case (K=3), as the number of data is increased, and this indicates that the kurtosis will converge to a finite single value if all moments of the distribution up to fourth are finite. The converged kurtosis value for the finite fourth-moment distributions and the number of data points needed to reach this value depend on the deviation of the original distribution from the Gaussian case.