The extremal family generated by the Yule process
The measurable space (\(\Omega\), \({\mathcal F})\) is considered, where \(\Omega\) denotes the set of all non-exploding, right-continuous step functions \(\omega:\;[0,\infty)\to \{0,1,...\}\) which are non-decreasing and skip free with \(\omega (0)=0\), and \({\mathcal F}\) denotes the \(\sigma\)-algebra \({\mathcal F}=\sigma (\cup_{t\geq 0}{\mathcal F}_ t)\) with \({\mathcal F}_ t\) the smallest \(\sigma\)-algebra of subsets of \(\Omega\) such that the mappings \(N_ s:\;\omega \to \omega (s)\) are measurable for every \(s\in [0,t]\). On (\(\Omega\), \({\mathcal F})\) the family \({\mathcal P}=\{P_{\alpha}\); \(\alpha\in (0,\infty)\}\) of probability measures \(P_{\alpha}\) is considered which are given by the stochastic (\({\mathcal F}_ t)\)- intensities \(\{\lambda_ t^{(\alpha)}\); \(t\geq 0\}\) with \(\lambda_ t^{(\alpha)}=\alpha (N_{t-}+1).\) Next the maximal family \({\mathcal M}\) generated by \({\mathcal P}\) is defined as the family of all probability measures P on (\(\Omega\), \({\mathcal F})\), such that for every \(t>0\), \[ P(B)=\int Q_{t,N_ t(\omega),X_ t(\omega)}(B)P(d\omega),\quad B\in {\mathcal F}_ t, \] holds, where \(X_ t=\int^{t}_{0}(N_ s+1)ds\) and \(Q_{t,n,x}\) denotes the conditional distribution \(P_{\alpha}(\cdot | N_ t=n\), \(X_ t=x)\) which does not depend on \(\alpha\). A minimal sufficient statistic for \({\mathcal P}\) restricted to \({\mathcal F}_ t\) is \((N_ t,X_ t)\). Moreover, the set \({\mathcal M}\), which is convex, can be seen as the largest family containing \({\mathcal P}\) for which \((N_ t,X_ t)\) is a sufficient statistic for all \(t\geq 0.\) The main result of the paper is a complete description of the family of extreme points of \({\mathcal M}\). This is done by determining the so-called Boltzmann family, i.e. the set of all probability measures P on (\(\Omega\), \({\mathcal F})\) such that, for some \(\omega\in \Omega\), the conditional distributions \(Q_{t,N_ t(\omega),X_ t(\omega)}\) weakly converge to P as \(t\to \infty\).
scientific article; zbMATH DE number 4153604
scientific article