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Poisson process Fock space representation, chaos expansion and covariance inequalities
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  • Published: 14 April 2010

Poisson process Fock space representation, chaos expansion and covariance inequalities

  • Günter Last1 &
  • Mathew D. Penrose2 

Probability Theory and Related Fields volume 150, pages 663–690 (2011)Cite this article

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Abstract

We consider a Poisson process η on an arbitrary measurable space with an arbitrary sigma-finite intensity measure. We establish an explicit Fock space representation of square integrable functions of η. As a consequence we identify explicitly, in terms of iterated difference operators, the integrands in the Wiener–Itô chaos expansion. We apply these results to extend well-known variance inequalities for homogeneous Poisson processes on the line to the general Poisson case. The Poincaré inequality is a special case. Further applications are covariance identities for Poisson processes on (strictly) ordered spaces and Harris–FKG-inequalities for monotone functions of η.

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References

  1. Barbour A.D., Holst L., Janson S.: Poisson Approximation. Clarendon Press, Oxford (1992)

    MATH  Google Scholar 

  2. Blaszczyszyn B.: Factorial-moment expansion for stochastic systems. Stoch. Proc. Appl. 56, 321–335 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bollobás B., Riordan O.: Percolation. Cambridge University Press, New York (2006)

    MATH  Google Scholar 

  4. Chen L.: Poincaré-type inequalities via stochastic integrals. Z. Wahrscheinlichkeitstheorie verw. Gebiete 69, 251–277 (1985)

    Article  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  6. Dellacherie, C., Meyer, P.A.: Probabilities and potential. In: Mathematics Studies, vol. 29. Hermann, Paris; North-Holland, Amsterdam (1978)

  7. Efron B., Stein C.: The jackknife estimate of variance. Ann. Stat. 9, 586–596 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  8. Georgii H., Küneth T.: Stochastic order of point processes. J. Appl. Prob. 34, 868–881 (1997)

    Article  MATH  Google Scholar 

  9. Heveling M., Reitzner M.: Poisson-Voronoi approximation. Ann. Appl. Probab. 19, 719–736 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hitsuda, M.: Formula for Brownian partial derivatives. In: Proceedings of the 2nd Japan-USSR Symposium on Probability Theory, pp. 111–114 (1972)

  11. Houdré C., Perez-Abreu V.: Covariance identities and inequalities for functionals on Wiener space and Poisson space. Ann. Probab. 23, 400–419 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Houdré C.: Remarks on deviation inequalities for functions of infinitely divisible random vectors. Ann. Probab. 30, 1223–1237 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Houdré C., Privault N.: Concentration and deviation inequalities in infinite dimensions via covariance representations. Bernoulli 8, 697–720 (2002)

    MathSciNet  MATH  Google Scholar 

  14. Itô K.: Multiple Wiener integral. J. Math. Soc. Jpn. 3, 157–169 (1951)

    Article  MATH  Google Scholar 

  15. Itô K.: Spectral type of the shift transformation of differential processes with stationary increments. Trans. Am. Math. Soc. 81, 253–263 (1956)

    Article  MATH  Google Scholar 

  16. Ito Y.: Generalized Poisson functionals. Probab. Theory Relat. Fields 77, 1–28 (1988)

    Article  MATH  Google Scholar 

  17. Kabanov Y.M.: On extended stochastic integrals. Theory Probab. Appl. 20, 710–722 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kabanov, Y.M., Skorokhod, A.V.: Extended stochastic integrals. In: Proceedings of the School-Seminar on the Theory of Random Processes, Part I. Druskininkai, November 25–30, 1974. Vilnius (Russian) (1975)

  19. Kallenberg O.: Random Measures. Akademie-Verlag, Berlin; Academic Press, London (1983)

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

    MATH  Google Scholar 

  21. Last, G., Penrose, M.D.: Martingale representation for Poisson processes with applications to minimal variance hedging. arXiv 1001.3972 (2010)

  22. Liebscher V.: On the isomorphism of Poisson space and symmetric Fock Space. In: Accardi, L. (eds) Quantum Probability and Related Topics IX, pp. 295–300. World Scientific, Singapore (1994)

    Google Scholar 

  23. Løkka A.: Martingale representation of functionals of Lévy processes. Stoch. Anal. Appl. 22, 867–892 (2005)

    Article  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Meester, R., Roy, R.: Continuum Percolation. Cambridge University Press (1996)

  27. Meyer P.A.: Quantum Probability for Probabilists, 2nd edn. Springer, Berlin (1995)

    MATH  Google Scholar 

  28. Møller J., Zuyev S.: Gamma-type results and other related properties of Poisson processes. Adv. Appl. Probab. 28, 662–673 (1996)

    Article  Google Scholar 

  29. Molchanov I., Zuyev S.: Variational analysis of functionals of Poisson processes. Math. Oper. Res. 25, 485–508 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  30. Nualart, D., Vives, J.: Anticipative calculus for the Poisson process based on the Fock space. In: Séminaire de Probabilités XXIV. Lecture Notes in Mathematics, vol. 1426, pp. 154–165 (1990)

  31. Ogura H.: Orthogonal functionals of the Poisson processes. Trans. IEEE Inf. Theory 18, 473–481 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  32. Peccati G., Solé J.L., Taqqu M.S., Utzet F.: Stein’s method and normal approximation of Poisson functionals. Ann. Probab. 38, 443–478 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Penrose M.: Random Geometric Graphs. Oxford University Press, Oxford (2003)

    Book  MATH  Google Scholar 

  34. Penrose M.D., Sudbury A.: Exact and approximate results for deposition and annihilation processes on graphs. Ann. Appl. Probab. 15, 853–889 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  35. Penrose M.D., Wade A.R.: Multivariate normal approximation in geometric probability. J. Stat. Theory Pract. 2, 293–326 (2008)

    MathSciNet  Google Scholar 

  36. Picard J.: On the existence of smooth densities for jump processes. Probab. Theory Relat. Fields 105, 481–511 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  37. Privault N.: Extended covariance identities and inequalities. Stat. Probab. Lett. 55, 247–255 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  38. Surgailis D.: On multiple Poisson stochastic integrals and associated Markov semigroups. Probab. Math. Stat. 3, 217–239 (1984)

    MathSciNet  Google Scholar 

  39. Skorohod A.V.: On a generalization of a stochastic integral. Theory Probab. Appl. 20, 219–233 (1975)

    Article  Google Scholar 

  40. Wiener N.: The homogeneous chaos. Am. J. Math. 60, 897–936 (1938)

    Article  MathSciNet  Google Scholar 

  41. Wu, L.: L 1 and modified logarithmic Sobolev inequalities and deviation inequalities for Poisson point processes. Preprint (1998)

  42. Wu L.: A new modified logarithmic Sobolev inequality for Poisson point processes and several applications. Probab. Theory Relat. Fields 118, 427–438 (2000)

    Article  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Karlsruhe Institute of Technology, 76128, Karlsruhe, Germany

    Günter Last

  2. Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK

    Mathew D. Penrose

Authors
  1. Günter Last
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  2. Mathew D. Penrose
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Corresponding author

Correspondence to Günter Last.

Additional information

Mathew D. Penrose partially supported by the Alexander von Humboldt Foundation through a Friedrich Wilhelm Bessel Research Award.

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Last, G., Penrose, M.D. Poisson process Fock space representation, chaos expansion and covariance inequalities. Probab. Theory Relat. Fields 150, 663–690 (2011). https://doi.org/10.1007/s00440-010-0288-5

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  • Received: 10 September 2009

  • Revised: 18 February 2010

  • Published: 14 April 2010

  • Issue Date: August 2011

  • DOI: https://doi.org/10.1007/s00440-010-0288-5

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Keywords

  • Poisson process
  • Chaos expansion
  • Derivative operator
  • Kabanov–Skorohod integral
  • Malliavin calculus
  • Poincaré inequality
  • Variance inequalities
  • Infinitely divisible random measure

Mathematics Subject Classification (2000)

  • 60G55
  • 60G51
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