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

WOS: 000365872100013PubMed ID: 26764857In 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.TUBITAK (Turkish Agency)Turkiye Bilimsel ve Teknolojik Arastirma Kurumu (TUBITAK) [112T083]We are indebted to Ayse Erzan for very fruitful discussions and useful remarks. This work has been supported by TUBITAK (Turkish Agency) under the Research Project No. 112T083