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تسجيل مستخدم جديدMass-structure of weighted real trees
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تأليف
Noah Forman
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Rooted, weighted continuum random trees are used to describe limits of sequences of random discrete trees. Formally, they are random quadruples $(mathcal{T},d,r,p)$, where $(mathcal{T},d)$ is a tree-like metric space, $rinmathcal{T}$ is a distinguished root, and $p$ is a probability measure on this space. The underlying branching structure is carried implicitly in the metric $d$. We explore various ways of describing the interaction between branching structure and mass in $(mathcal{T},d,r,p)$ in a way that depends on $d$ only by way of this branching structure. We introduce a notion of mass-structure equivalence and show that two rooted, weighted $mathbb{R}$-trees are equivalent in this sense if and only if the discrete hierarchies derived by i.i.d. sampling from their weights, in a manner analogous to Kingmans paintbox, have the same distribution. We introduce a family of trees, called interval partition trees that serve as representatives of mass-structure equivalence classes, and which naturally represent the laws of the aforementioned hierarchies.
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