Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T15:29:49.020Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE WEAK-HASH METRIC FOR BOUNDEDLY FINITE INTEGER-VALUED MEASURES

Part of: Classical measure theory Stochastic processes

Published online by Cambridge University Press:  19 July 2018

MAXIME MORARIU-PATRICHI Open the ORCID record for MAXIME MORARIU-PATRICHI [Opens in a new window]
MAXIME MORARIU-PATRICHI*
Affiliation:
Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK email m.morariu-patrichi14@imperial.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is known that the space of boundedly finite integer-valued measures on a complete separable metric space becomes a complete separable metric space when endowed with the weak-hash metric. It is also known that convergence under this topology can be characterised in a way that is similar to the weak convergence of totally finite measures. However, the original proofs of these two fundamental results assume that a certain term is monotonic, which is not the case as we show by a counterexample. We clarify these original proofs by addressing the parts that rely on this assumption and finding alternative arguments.

Keywords

boundedly finite integer-valued measuresweak-hash metriccompletenessseparabilityBorel sigma-algebra characterisationconvergence characterisation

MSC classification

Primary: 28A33: Spaces of measures, convergence of measures
Secondary: 60G55: Point processes
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 98 , Issue 2 , October 2018 , pp. 265 - 276
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

References

Daley, D. J. and Vere-Jones, D., An Introduction to the Theory of Point Processes, 2nd edn, Vol. I (Springer, New York, 2003).Google Scholar
Daley, D. J. and Vere-Jones, D., An Introduction to the Theory of Point Processes, 2nd edn, Vol. II (Springer, New York, 2008).Google Scholar
Kallenberg, O., Foundations of Modern Probability, 2nd edn (Springer, New York, 2002).Google Scholar
Kallenberg, O., Random Measures, Theory and Applications (Springer, Cham, 2017).Google Scholar
Matthes, K., Kerstan, J. and Mecke, J., Infinitely Divisible Point Processes (Akademie, Berlin, 1974).Google Scholar
Morariu-Patrichi, M. and Pakkanen, M. S., ‘Hybrid marked point processes: characterisation, existence and uniqueness’, Preprint, 2017, available at http://arxiv.org/abs/1707.06970.Google Scholar