Taylor's Law for some infinitely divisible probabbility distributions from population models

Abstract

In a family of random variables, Taylor's law or Taylor's power law offluctuation scaling is a variance function that gives the variance ${\sigma }^2>0$ of a random variable (rv) $X$ with expectation $\mu >0$ as a powerof $\mu$: ${\sigma }^2=A{\mu }^b$ for finite real $A>0,b$ that are thesame for all rvs in the family. Equivalently, TL holds when $\log {\sigma }^2=a+b\log \mu ,a=\log A$, for all rvs in some set. Here we analyze thepossible values of the TL exponent $b$ in five families of infinitelydivisible two-parameter distributions and show how the values of $b$ dependon the parameters of these distributions. The five families areTweedie-Bar-Lev-Enis, negative binomial, compound Poisson-geometric,compound geometric-Poisson (or Pólya-Aeppli), and gamma distributions.These families arise frequently in empirical data and population models, and they are limit laws of Markov processes that we exhibit in each case.