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

Area of Science:

• Statistical Mechanics
• Complex Systems Analysis
• Data Science

Background:

• A previous study proposed a universal power-law relation between skewness and kurtosis in complex dynamical systems.
• This relation suggested two non-Gaussian regimes with exponents 2 and 4/3, implying universality.

Purpose of the Study:

• To critically evaluate the universality of the proposed skewness-kurtosis relation.
• To investigate the influence of dataset size on this observed relation.
• To analyze the behavior of kurtosis for non-Gaussian distributions with increasing data.

Main Methods:

• Testing the proposed skewness-kurtosis relation using extensive synthetic datasets.
• Employing q-Gaussian distributions, a known family of non-Gaussian distributions.
• Analyzing the convergence of kurtosis as a function of dataset size.

Main Results:

• The purported universal relation between skewness and kurtosis was found to be an artifact of small datasets.
• This relation vanishes in sufficiently large datasets, particularly for q-Gaussians with finite fourth moments.
• Kurtosis was observed to saturate to a finite value (distinct from the Gaussian K=3) as data size increases.

Conclusions:

• The identified skewness-kurtosis relation is not a universal property of complex dynamical systems.
• The convergence of kurtosis to a finite value for large datasets indicates stability for distributions with finite fourth moments.
• The specific converged kurtosis value and data requirements depend on the initial deviation from Gaussianity.