Taylor’s Law for Some Infinitely Divisible Probability Distributions from Population Models

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