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

TL;DR

This paper investigates Taylor's law for various infinitely divisible distributions, analyzing how the fluctuation scaling exponent depends on distribution parameters within five key families relevant to population models.

Contribution

It characterizes the possible values of Taylor's law exponent for five important families of infinitely divisible distributions and links these values to distribution parameters.

Findings

Determines the range of Taylor's law exponent for each distribution family.

Shows how distribution parameters influence the fluctuation scaling.

Provides insights into empirical data modeling and population dynamics.

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 P\'{o}lya-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.