Taylor’s law for almost Poissonian processes

In this appendix, we prove the validity of Taylor’s law for processes driven by an uncorrelated sampling of a power-law frequency-rank distribution. To this end, we follow Gerlach and Altmann (2014). Let us imagine having a Poisson process for the occurrence of elements whose probabilities are defined in terms of a frequency-rank distribution |$f(R) \,{=}\, C R^{\alpha}$|⁠. We report here for convenience the expressions for normalisation factor |$C$| seen in Section 2.2: Let us not compute the expected value, |$\mu(N)$|⁠, and the variance, |$\sigma^2(N)$|⁠, of the number, |$D(N)$|⁠, of distinct elements that appeared in an uncorrelated sequence of length |$N$|⁠. For each element of rank |$R$|⁠, the probability that it appeared, at least once, in the sequence equals one minus the probability that the element never appeared in a sequence of |$N$| elements. This probability is thus given by |$1\,{-}\,e^{-N f(R)}$|⁠, i.e., corresponding to a Poisson process with frequency |$f(R)$|⁠. It has to be remarked that, though the process is Poissonian for each value of the rank |$R$|⁠, it is not overall a Poissonian process. The expected value, |$\mu(N)$|⁠, is thus given by: and the probability of zero occurrences is simply |$\exp(-f(R))$|⁠. For each element of rank |$R$|⁠, the probability that it appeared, at least once, in the sequence of |$N$| elements equals one minus the probability that the element never appeared in the sequence. This probability is thus given by |$1-e^{-N f(R)}$|⁠. It has to be remarked that, though the process is Poissonian for each value of the rank |$R$|⁠, it is not overall a Poissonian process. The expected value, |$\mu(N)$|⁠, is thus given by: It is useful to rewrite the previous equation by introducing the indicator-type function, i.e., |$I[x=0]=0$| and |$I[x \ge 0]=1$|⁠. Through this function, we can rewrite the previous equation in the following way: where |$n_R(N)$| is the number of occurrences of the element of rank |$R$| in the sequence of length |$N$|⁠. This formalism is very convenient to compute the variance, |$\sigma^2(N)$|⁠, of |$D(N)$|⁠, as: where we used that |$I[x]^2 \,{=}\, I[x]$| and the independency of Poisson processes corresponding to different elements (⁠|$R \ne R^{\prime})$|⁠. We also noticed that: If we now plug the expression |$f(R) \,{=}\, C R^{-\alpha}$|⁠, we can now compute the expression for |$\mu(N)$| and |$\sigma^2(N)$| as: By taking the continuum limit, one gets: