2025/06/20 by De Michele, Carlo, De Bartolo, Samuele
Economics, Econometrics and Finance · Physics and Astronomy · #Applications (stat.AP) #Complex Systems and Time Series Analysis #FOS: Computer and information sciences #FOS: Physical sciences #Mathematical Physics (math-ph) #Statistical Mechanics and Entropy #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2506.16906
openalex publication_date 2025/06/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Skewness and kurtosis are fundamental statistical moments commonly used to quantify asymmetry and tail behavior in probability distributions. Despite their widespread application in statistical mechanics, condensed matter physics, and complex systems, important aspects of their empirical behavior remain unclear, particularly in small samples and in relation to their hypothesized power law scaling. In this work, we address both issues using a combination of empirical and synthetic data. First, we establish a lower bound for sample kurtosis as a function of sample size and skewness. Second, we examine the conditions under which the 4/3 power law relationship between kurtosis and skewness emerges, effectively extending Taylor power law to higher order moments. Our results show that this scaling behavior predominantly occurs in data sampled from heavy tailed distributions and medium, large sample sizes.