Taylor's law, also known as fluctuation scaling in physics and the power-law variance function in statistics, is an empirical pattern widely observed across fields including ecology, physics, finance, and epidemiology. It states that the variance of a sample scales as a power function of the mean of the sample. We study generalizations of Taylor's law in the context of heavy-tailed distributions with infinite mean and variance. We establish the probabilistic limit and analyze the associated convergence rates. Our results extend the existing literature by relaxing the i.i.d. assumption to accommodate dependence and heterogeneity among the random variables. This generalization enables application to dependent data such as time series and network-structured data. We support the theoretical developments by extensive simulations, and the practical relevance through applications to real network data.

翻译：泰勒定律，在物理学中称为波动标度，在统计学中称为幂律方差函数，是一种在生态学、物理学、金融学和流行病学等多个领域广泛观测到的经验模式。它指出，样本的方差作为样本均值的幂函数进行标度。我们研究了在具有无限均值和方差的重尾分布背景下泰勒定律的推广。我们建立了概率极限并分析了相关的收敛速率。我们的结果通过放宽独立同分布假设，以容纳随机变量之间的依赖性和异质性，从而扩展了现有文献。这种推广使得该定律能够应用于时间序列和网络结构数据等依赖数据。我们通过大量的模拟支持了理论发展，并通过应用于真实网络数据证明了其实际相关性。