Marked metric measure spaces
Depperschmidt A, Greven A, Pfaffelhuber P (2011)
Publication Type: Journal article, Original article
Publication year: 2011
Journal
Publisher: Institute of Mathematical Statistics (IMS): OAJ / Institute of Mathematical Statistics
Book Volume: 16
Pages Range: 174-188
URI: https://projecteuclid.org/euclid.ecp/1465261973
Abstract
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 × 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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Germany (DE)How to cite
APA:
Depperschmidt, A., Greven, A., & Pfaffelhuber, P. (2011). Marked metric measure spaces. Electronic Communications in Probability, 16, 174-188.
MLA:
Depperschmidt, Andrej, Andreas Greven, and Peter Pfaffelhuber. "Marked metric measure spaces." Electronic Communications in Probability 16 (2011): 174-188.
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