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On the modified Palm version

Part of: Stochastic processes

Published online by Cambridge University Press:  01 February 2019

Hermann Thorisson
Hermann Thorisson*
Affiliation:
University of Iceland
*
Science Institute, University of Iceland, Dunhaga 5, 107 Reykjavik, Iceland. Email address: hermann@hi.is
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Abstract

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The interpretation of the ‘standard’ Palm version of a stationary random measure ξ is that it behaves like ξ conditioned on containing the origin in its mass. The interpretation of the ‘modified’ Palm version is that it behaves like ξ seen from a typical location in its mass. In this paper we shall focus on the modified Palm version, comparing it with the standard version in the transparent case of mixed biased coin tosses, and then establishing a limit theorem that motivates the above interpretation in the case of random measures on locally compact second countable Abelian groups possessing Følner averaging sets.

Keywords

Random measurePalm versionshift-couplingmass-stationarity

MSC classification

Primary: 60G57: Random measures 60G55: Point processes
Secondary: 60G60: Random fields
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue A: Branching and Applied Probability , December 2018 , pp. 271 - 280
Copyright
Copyright © Applied Probability Trust 2018 

References

[1]Aldous, D. J. and Thorisson, H. (1993).Shift-coupling.Stoch. Process. Appl. 44,1–14.Google Scholar
[2]Berbee, H. C. P. (1979).Random Walks with Stationary Increments and Renewal Theory (Math. Centre Tracts 112).Mathematisch Centrum,Amsterdam.Google Scholar
[3]Glynn, P. and Sigman, K. (1992).Uniform Cesàro limit theorems for synchronous processes with applications to queues.Stoch. Process. Appl. 40,29–43.Google Scholar
[4]Kallenberg, O. (2002).Foundations of Modern Probability,2nd edn.Springer,New York.Google Scholar
[5]Kallenberg, O. (2017).Random Measures, Theory and Applications.Springer,Cham.Google Scholar
[6]Last, G. (2010).Modern random measures: Palm theory and related models. In New Perspectives in Stochastic Geometry, eds W. Kendall and I. Molchanov,Oxford University Press, pp. 77–110.Google Scholar
[7]Last, G. and Thorisson, H. (2009).Invariant transports of stationary random measures and mass-stationarity.Ann. Prob. 37,790–813.Google Scholar
[8]Last, G. and Thorisson, H. (2011).What is typical? In New Frontiers in Applied Probability (Adv. Appl. Prob. Spec. Vol. 48A), eds C. M. Goldie and A. Mijatović, Applied Probability Trust,Sheffield, pp. 379–389.Google Scholar
[9]Last, G. and Thorisson, H. (2015).Construction and characterisation of stationary and mass-stationary random measures on ℝd.Stoch. Process. Appl. 125,4473–4488.Google Scholar
[10]Mecke, J. (1975).Invarianzeigenschaften allgemeiner Palmscher Maße.Math. Nachr. 65,335–344.Google Scholar
[11]Matthes, K., Kerstan, J. and Mecke, J. (1978).Infinitely Divisible Point Processes.John Wiley,Chichester.Google Scholar
[12]Nieuwenhuis, G. (1994).Bridging the gap between a stationary point process and its Palm distribution.Statist. Neerlandica 48,37–62.Google Scholar
[13]Palm, C. (1943).Intensitätsshwankungen im Fernsprechverkehr (Ericsson Techniks 44).Ericsson,Stockholm (in German). English translation: Intensity Variations in Telephone Traffic,North-Holland,Amsterdam, 1988.Google Scholar
[14]Slinyak, I. M. (1962).Some properties of stationary flows of homogeneous random events.Theory Prob. Appl. 7,336–341.Google Scholar
[15]Thorisson, H. (1995).On time- and cycle-stationarity.Stoch. Process. Appl. 55,183–209.Google Scholar
[16]Thorisson, H. (1996).Transforming random elements and shifting random fields.Ann. Prob. 24,2057–2064.Google Scholar
[17]Thorisson, H. (1999).Point-stationarity in d dimensions and Palm theory.Bernoulli 5,797–831.Google Scholar
[18]Thorisson, H. (2000).Coupling, Stationarity, and Regeneration.Springer,New York.Google Scholar