Skip to main content
SpringerLink
Log in

Normal Approximation of Poisson Functionals in Kolmogorov Distance

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

Peccati, Solè, Taqqu, and Utzet recently combined Stein’s method and Malliavin calculus to obtain a bound for the Wasserstein distance of a Poisson functional and a Gaussian random variable. Convergence in the Wasserstein distance always implies convergence in the Kolmogorov distance at a possibly weaker rate. But there are many examples of central limit theorems having the same rate for both distances. The aim of this paper was to show this behavior for a large class of Poisson functionals, namely so-called U-statistics of Poisson point processes. The technique used by Peccati et al. is modified to establish a similar bound for the Kolmogorov distance of a Poisson functional and a Gaussian random variable. This bound is evaluated for a U-statistic, and it is shown that the resulting expression is up to a constant the same as it is for the Wasserstein distance.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chen, L., Goldstein, L., Shao, Q.: Normal Approximation by Stein’s Method. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  2. Chen, L., Shao, Q.: Stein’s Method for Normal Approximation. An Introduction to Stein’s Method, pp. 1–59, Singapore University Press, Singapore (2005)

  3. Eichelsbacher, P., Thäle, C.: New Berry–Esseen Bounds for Non-Linear Functionals of Poisson Random Measures. arXiv:1310.1595 (2013)

  4. Lachièze-Rey, R., Peccati, G.: Fine Gaussian fluctuations on the Poisson space. I: contractions, cumulants and geometric random graphs. Electron. J. Probab. 18, Article 32 (2013)

  5. Lachièze-Rey, R., Peccati, G.: Fine Gaussian fluctuations on the Poisson space, II: rescaled kernels, marked processes and geometric U-statistics. Stoch. Process. Appl. 123, 4186–4218 (2013)

    Article  MATH  Google Scholar 

  6. Last, G.: Stochastic Analysis for Poisson Processes. arXiv:1405.4416 (2014)

  7. Last, G., Peccati, G., Schulte, M.: Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalitites and stabilization. arXiv:1401.7568 (2014)

  8. Last, G., Penrose, M.D.: Poisson process Fock space representation, chaos expansion and covariance inequalities. Probab. Theory Relat. Fields 150, 663–690 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Last, G., Penrose, M.D., Schulte, M., Thäle, C.: Moments and central limit theorems for some multivariate Poisson functionals. Adv. Appl. Probab. 46, 348–364 (2014)

    Article  MATH  Google Scholar 

  10. Nualart, D., Vives, J.: Anticipative Calculus for the Poisson Process Based on the Fock Space. Lecture Notes in Math. 1426, pp. 154–165 (1990)

  11. 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 

  12. Peccati, G., Taqqu, M.S.: Wiener Chaos: Moments, Cumulants and Diagram Formulae. Springer, Berlin (2011)

    Book  Google Scholar 

  13. Privault, N.: Stochastic Analysis in Discrete and Continuous Settings with Normal Martingales. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  14. Reitzner, M., Schulte, M.: Central limit theorems for U-statistics of Poisson point processes. Ann. Probab. 41, 3879–3909 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Reitzner, M., Schulte, M., Thäle, C.: Limit theory for the Gilbert graph. arXiv:1312.4861 (2013)

  16. Schulte, M., Thäle, C.: Distances between Poisson \(k\)-flats. Methodol. Comput. Appl. Probab. 16, 311–329 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Stein, C.: A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In: Proceedings of the Sixth Berkley Symposium on Mathematical Statistics Probability 2, University of California Press, Berkley (1972)

  18. Stein, C.: Approximate Computation of Expectations. Institute of Mathematical Statistics, Hayward, CA (1986)

    MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

Download references

Acknowledgments

Large parts of this paper were written during a stay at Case Western Reserve University (February to July 2012) supported by the German Academic Exchange Service. The author thanks Elizabeth Meckes, Giovanni Peccati, Matthias Reitzner, and Christoph Thäle for some valuable hints and helpful discussions.

Author information

Authors and Affiliations

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

    Matthias Schulte

Authors
  1. Matthias Schulte
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthias Schulte.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Schulte, M. Normal Approximation of Poisson Functionals in Kolmogorov Distance. J Theor Probab 29, 96–117 (2016). https://doi.org/10.1007/s10959-014-0576-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-014-0576-6

Keywords

Mathematics Subject Classification

Search

Navigation