Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-25T17:17:04.828Z Has data issue: false hasContentIssue false
  • English
  • Français

What is typical?

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Günter Last  and
Hermann Thorisson
Günter Last
Affiliation:
Karlsruhe Institute of Technology, Institut für Stochastik, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany
Hermann Thorisson
Affiliation:
University of Iceland, Science Institute, University of Iceland, Dunhaga 3, 107 Reykjavik, Iceland. Email address: hermann@hi.is
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.

Let ξ be a random measure on a locally compact second countable topological group, and let X be a random element in a measurable space on which the group acts. In the compact case we give a natural definition of the concept that the origin is a typical location for X in the mass of ξ, and prove that when this holds, the same is true on sets placed uniformly at random around the origin. This new result motivates an extension of the concept of typicality to the locally compact case where it coincides with the concept of mass-stationarity. We describe recent developments in Palm theory where these ideas play a central role.

Keywords

Random measuretypical locationPoisson processpoint-stationaritymass-stationarityPalm measureallocationinvariant transport

MSC classification

Primary: 60G57: Random measures 60G55: Point processes
Secondary: 60G60: Random fields
Type
Part 8. Point Processes
Information
Journal of Applied Probability , Volume 48 , Issue A: New Frontiers in Applied Probability (Journal of Applied Probability Special Volume 48A) , August 2011 , pp. 379 - 389
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Ferrari, P. A., Landim, C. and Thorisson, H., (2004). Poisson trees, succession lines and coalescing random walks. Ann. Inst. H. Poincaré Prob. Statist. 40, 141–152.CrossRefGoogle Scholar
[2] Heveling, M. and Last, G., (2005). Characterization of Palm measures via bijective point-shifts. Ann. Prob. 33, 1698–1715.CrossRefGoogle Scholar
[3] Heveling, M. and Last, G., (2007). Point shift characterization of Palm measures on abelian groups. Electron. J. Prob. 12, 122–137.CrossRefGoogle Scholar
[4] Holroyd, A. E. and Peres, Y., (2003). Trees and matchings from point processes. Electron. Commun. Prob. 8, 17–27.CrossRefGoogle Scholar
[5] Holroyd, A. E. and Peres, Y., (2005). Extra heads and invariant allocations. Ann. Prob. 33, 31–52.CrossRefGoogle Scholar
[6] Kallenberg, O., (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.CrossRefGoogle Scholar
[7] Last, G., (2010). Modern random measures: Palm theory and related models. In New Perspectives in Stochastic Geometry, eds Kendall, W. and Molchanov, I. Oxford University Press, pp. 77–110.Google Scholar
[8] Last, G. and Thorisson, H., (2009). Invariant transports of stationary random measures and mass-stationarity. Ann. Prob. 37, 790–813.CrossRefGoogle Scholar
[9] Last, G. and Thorisson, H., (2011). Characterization of mass-stationarity by Bernoulli and Cox transports. To appear in Commun. Stoch. Analysis.CrossRefGoogle Scholar
[10] Last, G. and Thorisson, H., (2011). Construction of stationary and mass-stationary random measures in ℝ d . In preparation.Google Scholar
[11] Mecke, J., (1967). Stationäre zufällige Mass e auf lokalkompakten Abelschen Gruppen. Z. Wahrscheinlichkeitsth. 9, 36–58.CrossRefGoogle Scholar
[12] Thorisson, H., (1999). Point-stationarity in d dimensions and Palm theory. Bernoulli 5, 797–831.CrossRefGoogle Scholar
[13] Thorisson, H., (2000). Coupling, Stationarity, and Regeneration. Springer, New York.CrossRefGoogle Scholar
[14] Timár, A., (2004). Tree and grid factors of general point processes. Electron. Commun. Prob. 9, 53–59.CrossRefGoogle Scholar