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2011 Marked metric measure spaces
Andrej Depperschmidt, Andreas Greven, Peter Pfaffelhuber
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Electron. Commun. Probab. 16: 174-188 (2011). DOI: 10.1214/ECP.v16-1615


A marked metric measure space (mmm-space) is a triple $(X,r,μ)$, where $(X,r)$ is a complete and separable metric space and $μ$ is a probability measure on $X \times I$ for some Polish space $I$ of possible marks. We study the space of all (equivalence classes of) marked metric measure spaces for some fixed $I$. It arises as a state space in the construction of Markov processes which take values in random graphs, e.g. tree-valued dynamics describing randomly evolving genealogical structures in population models. We derive here the topological properties of the space of mmm-spaces needed to study convergence in distribution of random mmm-spaces. Extending the notion of the Gromov-weak topology introduced in (Greven, Pfaffelhuber and Winter, 2009), we define the marked Gromov-weak topology, which turns the set of mmm-spaces into a Polish space. We give a characterization of tightness for families of distributions of random mmm-spaces and identify a convergence determining algebra of functions, called polynomials.


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Andrej Depperschmidt. Andreas Greven. Peter Pfaffelhuber. "Marked metric measure spaces." Electron. Commun. Probab. 16 174 - 188, 2011.


Accepted: 27 March 2011; Published: 2011
First available in Project Euclid: 7 June 2016

zbMATH: 1225.60009
MathSciNet: MR2783338
Digital Object Identifier: 10.1214/ECP.v16-1615

Primary: 05C80 , 60B10
Secondary: 60B05 , 60B12

Keywords: Gromov metric triples , Gromov- weak topology , metric measure space , population model , Prohorov metric

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