Correction to: Theoretical Ecology 13(1):1–5, 2020 https://doi.org/10.1007/s12080-019-00445-7

Dr. Thierry Huillet of CY Cergy Paris University (personal communication, November 25, 2021) kindly pointed out an error in the last line of Eq. (5) in my attempt to prove the portion of Theorem 1 pertaining to the quadratic generalization of Taylor’s law (QTL), Eq. (2), \(\mathrm{log}\;Var\left(X\left(p\right)\right)=\mathrm{log}\;a+{b}_{1}\mathrm{log}\;E(X\left(p\right))+{b}_{2}{\left(\mathrm{log}\;E(X\left(p\right))\right)}^{2}\). I made the same mistake in the last line of Eq. (8) in my attempt to prove a generalization of Theorem 1.

Happily, both Theorem 1 and the generalization of Theorem 1 remain true under additional conditions on the coefficients, which were omitted in the original statements. The claims follow immediately from Theorem 3, the proof of which is valid. Independent of Theorem 3, I give here a direct, elementary proof of Theorem 1 with additional conditions on \(a,{b}_{1}, {b}_{2}\) and P.

For real-valued random variables \(X, Y\), define \(X\sim Y\) if and only if, for some real \(c,d\ne 0,X\) has the same distribution as \(c + dY\).

Theorem 1

Let Z be any real-valued random variable with expectation \(EZ:=\mu \in \left(0,\infty \right)\) and variance \(Var\;Z:={\sigma }^{2}\in (0,\infty ).\) Let \(a,{b}_{1}, {b}_{2}\) be nonnegative real numbers such that \(a>0\) and at least one of \({b}_{1}, {b}_{2}\) is positive. If \({b}_{2}>0\), then there exists a family of random variables \(\{X(p)|p\in P\subset \left[{p}_{0},\infty \right), {p}_{0}>0\}\) such that the QTL holds for the chosen \(a,{b}_{1}, {b}_{2}\) and \(X\left(p\right)\sim Z\) for every \(p\in P\). If  \({b}_{1}>0, {b}_{2}=0\), then TL, namely, \(\mathrm{log}\;Var\left(X\left(p\right)\right)=\mathrm{log}\;a+{b}_{1}\mathrm{log}\;E(X\left(p\right))\), holds.

Proof

Given \(a,{b}_{1}, {b}_{2}\), pick a positive number g that is large enough to guarantee that \(\mathrm{log}\;a+{b}_{1}gp+{b}_{2}{g}^{2}{p}^{2}>0\) for every \(p\in P\). Such a choice is possible because of our assumptions about \(a,{b}_{1}, {b}_{2}\) and P. Define, for every \(p\in P\),

$$X\left(p\right):={e}^{gp}+\sqrt{\mathrm{log}\;a+{b}_{1}gp+{b}_{2}{g}^{2}{p}^{2}}(Z-\mu )/\sigma$$
(1)

Then for each \(p\in P\), \(X\left(p\right)\) is a linear function of \(Z\), so \(X\left(p\right)\sim Z\), \(EX\left(p\right)={e}^{gp}\), \(\mathrm{log}\;EX\left(p\right)=gp\ne 0\), and the QTL holds because

$$\begin{aligned}Var X\left(p\right)&=\mathrm{log}a+{b}_{1}gp+{b}_{2}{g}^{2}{p}^{2}\\&=\mathrm{log}a+{b}_{1}\mathrm{log}E(X\left(p\right))+{b}_{2}{\left(\mathrm{log}E(X\left(p\right))\right)}^{2}\end{aligned}$$
(2)

A polynomial Taylor’s law generalizes Theorem 1. The variance function

$$\mathrm{log}\;Var\left(X\left(p\right)\right)=\mathrm{log}\;a+\sum_{i=1}^{\infty }{b}_{i}{\left[\mathrm{log}\;E(X\left(p\right))\right]}^{i}$$
(3)

follows similarly by defining

$$X\left(p\right):={e}^{gp}+\sqrt{\mathrm{log}\;a+\sum_{i=1}^{\infty }{b}_{i}{\left[gp\right]}^{i}}(Z-\mu )/\sigma$$
(4)

under conditions on \(a,{b}_{i},\) and P sufficient to guarantee that \(\mathrm{log}\;a+\sum_{i=1}^{\infty }{b}_{i}{\left[gp\right]}^{i}>0\).

The affiliations in the original publication should be replaced with: Laboratory of Populations, Rockefeller University and Columbia University, 1230 York Avenue, Box 20, New York, NY 10065, USA; Earth Institute and Department of Statistics, Columbia University, New York, NY 10027, USA; and Department of Statistics, University of Chicago, Chicago, IL 60637, USA.