Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Invariant Palm and related disintegrations via skew factorization
Download PDF
Download PDF
  • Published: 19 November 2009

Invariant Palm and related disintegrations via skew factorization

  • Olav Kallenberg1 

Probability Theory and Related Fields volume 149, pages 279–301 (2011)Cite this article

  • 151 Accesses

  • 4 Citations

  • Metrics details

Abstract

Let G be a measurable group with Haar measure λ, acting properly on a space S and measurably on a space T. Then any σ-finite, jointly invariant measure M on S × T admits a disintegration \({\nu \otimes \mu}\) into an invariant measure ν on S and an invariant kernel μ from S to T. Here we construct ν and μ by a general skew factorization, which extends an approach by Rother and Zähle for homogeneous spaces S over G. This leads to easy extensions of some classical propositions for invariant disintegration, previously known in the homogeneous case. The results are applied to the Palm measures of jointly stationary pairs (ξ, η), where ξ is a random measure on S and η is a random element in T.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Daley D.J., Vere-Jones D.: An Introduction to the Theory of Point Processes II, 2nd edn. Springer, New York (2008)

    Book  MATH  Google Scholar 

  2. Dellacherie C., Meyer P.A.: Probabilités et Potentiel, Chap. I–IV. Hermann, Paris (1975)

    Google Scholar 

  3. Dudley R.M.: Real Analysis and Probability. Wadsworth & Brooks/Cole, Belmont (1989)

    MATH  Google Scholar 

  4. Geman D., Horowitz J.: Remarks on Palm measures. Ann. Inst. H. Poincaré Sec. B 9, 215–232 (1973)

    MATH  MathSciNet  Google Scholar 

  5. Geman D., Horowitz J.: Random shifts which preserve measure. Proc. Am. Math. Soc. 49, 143–150 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  6. Gentner, D., Last, G.: Palm pairs and the general mass transport principle. Math. Z. (2009, in press)

  7. Harris T.E.: Random measures and motions of point processes. Z. Wahrschein. verw. Geb. 18, 85–115 (1971)

    Article  MATH  Google Scholar 

  8. Hewitt E., Ross K.A.: Abstract Harmonic Analysis I, 2nd edn. Springer, New York (1979)

    Google Scholar 

  9. Holroyd A.E., Peres Y.: Extra heads and invariant allocations. Ann. Probab. 33, 31–52 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kallenberg O.: Random Measures, 4th edn. Akademie-Verlag/Academic Press, Berlin/London (1986)

    Google Scholar 

  11. Kallenberg O.: Foundations of Modern Probability, 2nd edn. Springer, New York (2002)

    MATH  Google Scholar 

  12. Kallenberg O. Invariant measures and disintegrations with applications to Palm and related kernels. Probab. Theory Relat. Fields 139, 285–310, 311 (2007)

  13. Krickeberg K.: Invariance properties of the correlation measure of line-processes. In: Harding, E.F., Kendall, D.G. (eds) Stochastic Geometry, pp. 76–88. Wiley, London (1974)

    Google Scholar 

  14. Krickeberg K.: Moments of point-processes. In: Harding, E.F., Kendall, D.G. (eds) Stochastic Geometry, pp. 89–113. Wiley, London (1974)

    Google Scholar 

  15. Last, G.: Modern random measures: Palm theory and related models. In: Kendall, W., Molchanov, I. (eds.) New perspectives in Stochastic Geometry. Oxford University Press, Oxford (2009, in press)

  16. Last, G.: Stationary random measures on homogeneous spaces. J. Theor. Probab. (2009, in press)

  17. Last G., Thorisson H.: Invariant transports of stationary random measures and mass-stationarity. Ann. Probab. 37, 790–813 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. Matthes K.: Stationäre zufällige Punktfolgen I. J.-ber. Deutsch. Math.-Verein. 66, 66–79 (1963)

    MathSciNet  Google Scholar 

  19. Matthes K., Kerstan J., Mecke J.: Infinitely Divisible Point Processes. Wiley, Chichester (1978)

    MATH  Google Scholar 

  20. Mecke J.: Stationäre zufällige Maße auf lokalkompakten Abelschen Gruppen. Z. Wahrsch. verw. Geb. 9, 36–58 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  21. Mecke J.: Invarianzeigenschaften allgemeiner Palmscher Maße. Math. Nachr. 65, 335–344 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  22. Neveu J.: Sur les mesures de Palm de deux processus ponctuels stationnaires. Z. Wahrschein. verw. Geb. 34, 199–203 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  23. Neveu, J.: Processus ponctuels. In: École d’été de probabilités de Saint-Flour VI–1976. Lecture Notes in Mathematics, vol. 598, pp. 249–445. Springer, Berlin (1977)

  24. Papangelou F.: On the Palm probabilities of processes of points and processes of lines. In: Harding, E.F., Kendall, D.G. (eds) Stochastic Geometry, pp. 114–147. Wiley, London (1974)

    Google Scholar 

  25. Port S.C., Stone C.J.: Infinite particle systems. Trans. Am. Math. Soc. 178, 307–340 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  26. Rother W., Zähle M.: Palm distributions in homogeneous spaces. Math. Nachr. 149, 255–263 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  27. Ryll-Nardzewski, C.: Remarks on processes of calls. In: Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, vol. 2, pp. 455–465 (1961)

  28. Stoyan D., Kendall W.S., Mecke J.: Stochastic Geometry and its Applications, 2nd edn. Wiley, Chichester (1995)

    MATH  Google Scholar 

  29. Thorisson H.: Coupling, Stationarity, and Regeneration. Springer, New York (2000)

    MATH  Google Scholar 

  30. Tortrat A.: Sur les mesures aléatoires dans les groupes non abéliens. Ann. Inst. H. Poincaré Sec. B 5, 31–47 (1969)

    MATH  MathSciNet  Google Scholar 

  31. Weil A.: La mesure invariante dans les espaces de groupes et les espaces homogènes. Enseignement Math. 35, 241 (1936)

    Google Scholar 

  32. Weil, A.: L’intégration dans les groupes topologiques et ses applications. Actualités Sci. et Ind., vol. 869, p. 1145. Hermann et Cie, Paris (1940)

  33. Zähle U.: Self-similar random measures I. Notion, carrying Hausdorff dimension, and hyperbolic distribution. Probab. Theory Relat. Fields 80, 79–100 (1988)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Auburn University, 221 Parker Hall, Auburn, AL, 36849, USA

    Olav Kallenberg

Authors
  1. Olav Kallenberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Olav Kallenberg.

Additional information

When this work was completed, I became aware of the paper [6] on a related subject, written independently of the present one and finished at about the same time. Though some overlap is inevitable (in particular, versions of our Theorem 5.1 appear in both papers), the approach and emphasis are different, and the interested reader may want to study both papers in parallel.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kallenberg, O. Invariant Palm and related disintegrations via skew factorization. Probab. Theory Relat. Fields 149, 279–301 (2011). https://doi.org/10.1007/s00440-009-0254-2

Download citation

  • Received: 20 January 2009

  • Revised: 05 October 2009

  • Published: 19 November 2009

  • Issue Date: February 2011

  • DOI: https://doi.org/10.1007/s00440-009-0254-2

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Invariant disintegration
  • Orbit selection
  • Inversion kernel
  • Skew factorization
  • Duality
  • Stationary random measures
  • Palm kernel

Mathematics Subject Classification (2000)

  • 28C10
  • 60G57
  • 60G10
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature