The gap between Gromov-vague and Gromov-Hausdorff-vague topology

Athreya, Siva ; Lohr, Wolfgang ; Winter, Anita (2016) The gap between Gromov-vague and Gromov-Hausdorff-vague topology Stochastic Processes and their Applications, 126 (9). pp. 2527-2553. ISSN 0304-4149

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In Athreya et al. (2015) an invariance principle is stated for a class of strong Markov processes on tree-like metric measure spaces. It is shown that if the underlying spaces converge Gromov vaguely, then the processes converge in the sense of finite dimensional distributions. Further, if the underlying spaces converge Gromov-Hausdorff vaguely, then the processes converge weakly in path space. In this paper we systematically introduce and study the Gromov-vague and the Gromov-Hausdorff-vague topology on the space of equivalence classes of metric boundedly finite measure spaces. The latter topology is closely related to the Gromov-Hausdorff-Prohorov metric which is defined on different equivalence classes of metric measure spaces. We explain the necessity of these two topologies via several examples, and close the gap between them. That is, we show that convergence in Gromov-vague topology implies convergence in Gromov-Hausdorff-vague topology if and only if the so-called lower mass-bound property is satisfied. Furthermore, we prove and disprove Polishness of several spaces of metric measure spaces in the topologies mentioned above. As an application, we consider the Galton-Watson tree with critical offspring distribution of finite variance conditioned to not get extinct, and construct the so-called Kallenberg-Kesten tree as the weak limit in Gromov-Hausdorff-vague topology when the edge length is scaled down to go to zero.

Item Type:Article
Source:Copyright of this article belongs to Elsevier Science.
Keywords:Metric Measure Spaces; Gromov-vague Topology; Gromov-Hausdorff-vague; Gromov-weak; Gromov-Hausdorff-weak; Gromov-Prohorov Metric; Lower Mass-bound Property; Full Support Assumption; Coding Trees By Excursions; Kallenberg Tree
ID Code:99910
Deposited On:12 Feb 2018 12:16
Last Modified:12 Feb 2018 12:16

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