Generalized Taylor's law for infinite-mean heavy-tailed data under dependence and heterogeneity.

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 some power 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 assumption that the observed random variables are independently and identically distributed to accommodate dependence and heterogeneity among them. This generalization enables application of Taylor's law to dependent time series and network-structured data. We support the theoretical developments by extensive simulations, and the practical relevance through applications to real network data.