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

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Abstract

In a family of random variables, Taylor’s law or Taylor’s power law of fluctuation scaling is a variance function that gives the variance \(\sigma ^{2}>0\) of a random variable (rv) X with expectation \(\mu >0\) as a power of \(\mu \): \(\sigma ^{2}=A\mu ^{b}\) for finite real \(A>0,\ b\) that are the same 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 the possible values of the TL exponent b in five families of infinitely divisible two-parameter distributions and show how the values of b depend on the parameters of these distributions. The five families are Tweedie–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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Acknowledgements

We thank two reviewers for excellent constructive comments. T.H. acknowledges partial support from the “Chaire Modélisation mathématique et biodiversité” of Veolia-Ecole Polytechnique-MNHN-Fondation X, and support from the labex MME-DII Center of Excellence (Modèles mathématiques et économiques de la dynamique, de l’incertitude et des interactions, ANR-11-LABX-0023-01 project). This work was also funded by CY Initiative of Excellence (Grant “ Investissements d’Avenir” ANR- 16-IDEX-0008), Project “EcoDep” PSI-AAP2020-0000000013. The ECODEP project is organized by Paul Doukhan. We are grateful for this opportunity to collaborate.

Author information

Authors and Affiliations

  1. Laboratory of Populations, Rockefeller University and Columbia University, 1230 York Avenue, Box 20, New York, NY, 10065, USA

    Joel E. Cohen

  2. Earth Institute and Department of Statistics, Columbia University, New York, 10027, USA

    Joel E. Cohen

  3. Department of Statistics, University of Chicago, Chicago, IL, 60637, USA

    Joel E. Cohen

  4. Laboratoire de Physique Théorique et Modélisation, CY Cergy Paris University, CNRS UMR-8089, Site de Saint Martin, 2 Avenue Adolphe-Chauvin, 95302, Cergy-Pontoise, France

    Thierry E. Huillet

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Correspondence to Joel E. Cohen.

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Communicated by Li-Cheng Tsai.

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Appendix

Appendix

Here we briefly sketch that if X is SD, it is a weak limit of a pure-death branching process with immigration (if \(\mathbb {N}_{0}\)-valued), or a weak limit of a continuous-time Lévy-driven Ornstein-Uhlenbeck process (if continuous). The analysis concerns the rvs X obeying TL with \(a=0\). From the scaling transform (14) introducing \(a=\log \sigma _{1}^{2}\) , the modifications for \(a\ne 0\) could be readily obtained. In both discrete and continuous cases, a population that is randomly annihilated is randomly regenerated by the recurrent arrivals of random quantities of immigrants, yielding a stationary or invariant distribution of population size.

1.1 Discrete Self-decomposable rvs and Pure-Death Branching Processes with Immigration in Continuous Time

van Harn et al. [33] construct a regenerative process in continuous time that produces discrete SD distributions in the long run. Consider a continuous-time homogeneous compound Poisson process \(P_{r}(t)\), \(t\ge 0\), \( P_{r}(0)=0\), having rate \(r>0\), with pgf

$$\begin{aligned} {E}_{P_{r}(0)=0}(z^{P_{r}(t)})=\exp \{-rt(1-h(z))\}, \end{aligned}$$
(15)

where h(z), with \(h(0)=0\), is the pgf of the sizes of the clones or immigrant clusters arriving at the jump times of \(P_{r}(t)\). Let

$$\begin{aligned} \varphi _{t}(z)=1-e^{-t}(1-z) \end{aligned}$$
(16)

be the pgf of a pure-death branching process started with one particle at \( t=0\). (More general subcritical branching processes could be considered.) This expression of \(\varphi _{t}(z)\) is easily seen to solve \(\overset{.}{ \varphi }_{t}(z)=f(\varphi _{t}(z))=1-\varphi _{t}(z)\), \(\varphi _{0}(z)=z\), as is usual for a pure-death continuous-time Bellman-Harris branching process [21] with affine branching mechanism \(f(z)=r_{d}(1-z)\) with fixed death rate \(r_{d}=1\). The distribution function of the lifetime of the initial particle is \(1-e^{-t}\). Let \(X_{t}\) with \(X_{0}=0\) be a random process counting the current size of some population for which a random number of individuals (determined by h(z)) immigrate at the jump times of \(P_{r}(t)\). Each newly arrived individual is independently and immediately subject to the pure death process above. We have

$$\begin{aligned} \phi _{t}(z):={E}(z^{X_{t}})=\exp \left\{ -r\int \limits _{0}^{t}\left[ 1-h(\varphi _{s}(z))\right] ds\right\} ,\quad \phi _{0}(z)=1, \end{aligned}$$
(17)

with \(\phi _{t}(0)={P}(X_{t}=0)=\exp \{-r\int _{0}^{t}(1-h(1-e^{-s}))ds\},\) the probability that the population is extinct at t. As \(t\rightarrow \infty \),

$$\begin{aligned} \phi _{t}(z)\rightarrow \phi _{\infty }(z)&=\exp \left\{ -r\int \limits _{0}^{\infty }\left[ 1-h(1-e^{-s}(1-z))\right] ds\right\} \nonumber \\&=\exp \left\{ -r\int \limits _{z}^{1}\frac{1-h\left( u\right) }{1-u}du\right\} . \end{aligned}$$
(18)

So \(X:=X_{\infty },\) as the limiting population size of this pure-death process with immigration, is a SD rv [33]. Define the rv \(X_{c}\) implicitly by requiring that \(X\overset{d}{=}cX^{\prime }+X_{c}\), where \( X^{\prime }\) is an iid copy of X and \(0<c<1\). Then

$$\begin{aligned} \phi _{X_{c}}(z)=\frac{\phi _{\infty }(z)}{\phi _{\infty }(1-c(1-z))}=\exp \left\{ -r\int \limits _{z}^{1-c(1-z)}\frac{1-h(u)}{1-u}du\right\} \end{aligned}$$

is a pgf. In such models typically, a decaying subcritical branching population is regenerated by a random number of incoming immigrants at random Poissonian times.

1.2 Continuous Self-decomposable rvs and Ornstein–Uhlenbeck Process in Continuous-Time

When X is continuous and SD, X is the limiting distribution of population size as \(t\rightarrow \infty \) of some Ornstein-Uhlenbeck process \(X_{t}\):

$$\begin{aligned} dX_{t}=-X_{t}dt+d\mathcal {L}_{t}, \ \ X_{0}=0, \end{aligned}$$

driven by the Lévy process \(\mathcal {L}_{t}\) for which

$$\begin{aligned} Ee^{-\lambda \mathcal {L}_{t}}=\Phi _{0}( \lambda ) ^{t}, \ \ t\ge 0, \end{aligned}$$

where \(\Phi _{0}( \lambda ) \) is the PLSt of an infinitely divisible rv appearing in the representation (7) of \(\Phi ( \lambda ) =Ee^{-\lambda X}\). See [26].

We now show that for a TweBLE rv with \(\alpha \in ( -\infty ,0) \), there is no \(L_{0}( \lambda ) =-\log \Phi _{0}( \lambda ) \) such that \(L_{0}^{\prime }( \lambda ) \) is completely monotone on \(( 0,\infty ) \). This result means that the TweBLE rv for \(\alpha \in ( -\infty ,0) \) is not SD, just infinitely divisible. Indeed, with \(L(\lambda )=-\log \Phi (\lambda )\),

$$\begin{aligned} L_{0}(\lambda )= & {} \lambda L^{\prime }(\lambda )=(1-\alpha )^{1-\alpha }\lambda (\theta +\lambda )^{\alpha -1}, \\ L_{0}^{\prime }(\lambda )= & {} (1-\alpha )^{1-\alpha }(\theta +\lambda )^{\alpha -2}[\theta +\lambda \alpha ], \end{aligned}$$

with \(L_{0}^{\prime }(\lambda )>0\) only if \(\lambda>\lambda _{c}=-\theta /\alpha >0,\) so not in the full range \(\lambda \in (0,\infty )\). So \(\Phi _{0}(\lambda )\) is not completely monotone on \((0,\infty )\), and is therefore not an infinitely divisible PLSt. This Poisson-gamma regime for which the limiting distribution is a Poisson sum P of iid gamma-distributed clusters of size \(\Delta \) was studied by [19, p. 17, Sect. 3.3.2], who underline what they call its “impact inhomogeneity”: \(E(\Delta )=C(\alpha )\cdot E(P)^{-1/\alpha }\) for some constant \(C(\alpha )>0\).

By contrast, if \(\alpha \in (0,1)\), then \(L_{0}(\lambda )\) may be written as

$$\begin{aligned} L_{0}(\lambda )=\int \limits _{0}^{\infty }(1-e^{-\lambda x})\pi (x)dx \end{aligned}$$

where

$$\begin{aligned} \pi (x)dx=\frac{1}{\Gamma (1-\alpha )}x^{-(\alpha +1)}(\alpha +\theta x)e^{-\theta x}dx \end{aligned}$$

is a tempered Lévy measure integrating \(1\wedge x\). The driving process \(\mathcal {L}_{t}\) of \(X_{t}\), with PLSt \(Ee^{-\lambda \mathcal {L} _{t}}=e^{-tL_{0}(\lambda )}\), is a subordinator and \(X=X_{\infty }\) is a SD TweBLE rv obtained as the limiting distribution of the corresponding Ornstein-Uhlenbeck process. One could extend this construction to other SD subordinated Lévy families, such as those in [1, 6, 10, 28, 29].

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Cohen, J.E., Huillet, T.E. Taylor’s Law for Some Infinitely Divisible Probability Distributions from Population Models. J Stat Phys 188, 33 (2022). https://doi.org/10.1007/s10955-022-02962-y

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