Electronic Journal of Probability

Point shift characterization of Palm measures on Abelian groups

Matthias Heveling and Gunter Last

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Our first aim in this paper is to characterize Palm measures of stationary point processes through point stationarity. This generalizes earlier results from the Euclidean case to the case of an Abelian group. While a stationary point process looks statistically the same from each site, a point stationary point process looks statistically the same from each of its points. Even in the Euclidean case our proof will simplify some of the earlier arguments. A new technical result of some independent interest is the existence of a complete countable family of matchings. Using a change of measure we will generalize our results to discrete random measures. In the Euclidean case we will finally treat general random measures by means of a suitable approximation.

Article information

Electron. J. Probab., Volume 12 (2007), paper no. 5, 122-137.

Accepted: 4 February 2007
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G55: Point processes
Secondary: 60G57: Random measures

point process random measure stationarity point-stationarity Palm measure matching bijective point map

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Heveling, Matthias; Last, Gunter. Point shift characterization of Palm measures on Abelian groups. Electron. J. Probab. 12 (2007), paper no. 5, 122--137. doi:10.1214/EJP.v12-394. https://projecteuclid.org/euclid.ejp/1464818476

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  • Daley, D. J.; Vere-Jones, D. An introduction to the theory of point processes. Springer Series in Statistics. Springer-Verlag, New York, 1988. xxii+702 pp. ISBN: 0-387-96666-8
  • Heveling, Matthias; Last, Günter. Characterization of Palm measures via bijective point-shifts. Ann. Probab. 33 (2005), no. 5, 1698–1715.
  • Holroyd, Alexander E.; Peres, Yuval. Trees and matchings from point processes. Electron. Comm. Probab. 8 (2003), 17–27 (electronic).
  • Kallenberg, Olav. Foundations of modern probability. Probability and its Applications (New York). Springer-Verlag, New York, 2002. xx+638 pp. ISBN: 0-387-95313-2
  • Last, Günter. Stationary partitions and Palm probabilities. Adv. in Appl. Probab. 38 (2006), no. 3, 602–620.
  • Matthes, Klaus. Stationäre zufällige Punktfolgen. I. (German) Jber. Deutsch. Math.-Verein 66 1963/1964 Abt. 1 66–79.
  • Matthes, Klaus; Kerstan, Johannes; Mecke, Joseph. Infinitely divisible point processes. John Wiley & Sons, Chichester-New York-Brisbane, 1978. xii+532 pp. ISBN: 0-471-99460-X
  • Mecke, J. Stationäre zufällige Masse auf lokalkompakten Abelschen (German) Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 9 1967 36–58.
  • Mecke, J. Invarianzeigenschaften allgemeiner Palmscher Maße. (German) Math. Nachr. 65 (1975), 335–344.
  • Neveu, J. Processus ponctuels. pp. 249–445. Lecture Notes in Math., Vol. 598, Springer-Verlag, Berlin, 1977.
  • Thorisson, Hermann. Coupling, stationarity, and regeneration. Probability and its Applications (New York). Springer-Verlag, New York, 2000. xiv+517 pp. ISBN: 0-387-98779-7
  • Timár, Ádám. Tree and grid factors for general point processes. Electron. Comm. Probab. 9 (2004), 53–59 (electronic).