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

In a family of random variables, Taylor's law or Taylor's power law offluctuation scaling is a variance function that gives the variance $σ^{2}>0$ of a random variable (rv) $X$ with expectation $μ>0$ as a powerof $μ$: $σ^{2}=Aμ^{b}$ for finite real $A>0,\ b$ that are thesame for all rvs in the family. Equivalently, TL holds when $\log σ^{2}=a+b\log μ,\ a=\log A$, 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.

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