Results for the Taylor’s law in the Extended Urn Model with Triggering

In Section 8.1.3 of the main text, we introduced the representation of the Extended Urn Model with Triggering (EUMT) in terms of a triangular two-colour scheme and defined the so-called triangular replacement matrix (Eq. 8.22). Thanks to that representation, it is possible to exploit results on triangular urn models, as those derived in Janson (2004, 2005) and Aguech (2009) to derive the analytic expression of Taylor’s law for the EUMT models. We presented the results in the main text. Though the full derivation of those is outside the scope of this book, we give here a sketch of how to derive the main results (see also Tria et al., 2020). We present the results organised in different cases, depending on the values of the parameters, |$\nu$| and |$\rho$|⁠. In this case, one can prove the following result: where |$Z$| is a suitable random variable with finite moments, and |$\stackrel{a.s.}\longrightarrow$| means almost sure convergence. In the particular case of |$a\,{=}\,0$|⁠, we can write an explicit expression for all the moments of |$Z$| as: Those results are also reported in Pitman (2006) for the two-parameters Poisson–Dirichlet process, with |$\theta\,{=}\,N_0/\rho$| and |$\alpha\,{=}\,\nu/\rho$| (see Section 8.6 of this book). For |$a \,{\neq}\, 0$|⁠, the probability density function and its moments are given in implicit form Janson (2005).