  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.
