Lectures on Gaussian Approximations with Malliavin Calculus

Chapter
First Online:
Part of the Lecture Notes in Mathematics book series (LNM, volume 2078)

Overview.

In a seminal paper of 2005, Nualart and Peccati [40] discovered a surprising central limit theorem (called the “Fourth Moment Theorem” in the sequel) for sequences of multiple stochastic integrals of a fixed order: in this context, convergence in distribution to the standard normal law is equivalent to convergence of just the fourth moment. Shortly afterwards, Peccati and Tudor [46] gave a multidimensional version of this characterization.

Keywords

Brownian Motion Fractional Brownian Motion Hermite Polynomial Fourth Moment Free Probability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Notes

Acknowledgements

It is a pleasure to thank the Fondation des Sciences Mathématiques de Paris for its generous support during the academic year 2011–2012 and for giving me the opportunity to speak about my recent research in the prestigious Collège de France. I am grateful to all the participants of these lectures for their assiduity. Also, I would like to warmly thank two anonymous referees for their very careful reading and for their valuable suggestions and remarks. Finally, my last thank goes to Giovanni Peccati, not only for accepting to give a lecture (resulting to the material developed in Sect. 10) but also (and especially!) for all the nice theorems we recently discovered together. I do hope it will continue this way as long as possible!

References

  1. 1.
    B. Bercu, I. Nourdin, M.S. Taqqu, Almost sure central limit theorems on the Wiener space. Stoch. Proc. Appl. 120(9), 1607–1628 (2010)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    P. Biane, R. Speicher, Stochastic analysis with respect to free Brownian motion and analysis on Wigner space. Probab. Theory Rel. Fields 112, 373–409 (1998)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    H. Biermé, A. Bonami, J. Léon, Central limit theorems and quadratic variations in terms of spectral density. Electron. J. Probab. 16, 362–395 (2011)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    H. Biermé, A. Bonami, I. Nourdin, G. Peccati, Optimal Berry-Esseen rates on the Wiener space: the barrier of third and fourth cumulants. ALEA Lat. Am. J. Probab. Math. Stat. 9(2), 473–500 (2012)Google Scholar
  5. 5.
    E. Bolthausen, An estimate of the remainder in a combinatorial central limit theorem. Z. Wahrscheinlichkeitstheorie verw. Gebiete 66, 379–386 (1984)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    J.-C. Breton, I. Nourdin, Error bounds on the non-normal approximation of Hermite power variations of fractional Brownian motion. Electron. Comm. Probab. 13, 482–493 (2008) (electronic)Google Scholar
  7. 7.
    P. Breuer, P. Major, Central limit theorems for non-linear functionals of Gaussian fields. J. Mult. Anal. 13, 425–441 (1983)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    L.H.Y. Chen, Poisson approximation for dependent trials. Ann. Probab. 3(3), 534–545 (1975)MATHCrossRefGoogle Scholar
  9. 9.
    L.H.Y. Chen, L. Goldstein, Q.-M. Shao, Normal Approximation by Stein’s Method. Probability and Its Applications. Springer, New York (2010)Google Scholar
  10. 10.
    F. Daly, Upper bounds for Stein-type operators. Electron. J. Probab. 13(20), 566–587 (2008) (electronic)Google Scholar
  11. 11.
    L. Decreusefond, D. Nualart, Hitting times for Gaussian processes. Ann. Probab. 36(1), 319–330 (2008)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    A. Deya, S. Noreddine, I. Nourdin, Fourth moment theorem and q-Brownian chaos. Commun. Math. Phys. 1–22 (2012)Google Scholar
  13. 13.
    A. Deya, I. Nourdin, Invariance principles for homogeneous sums of free random variables. arXiv preprint arXiv:1201.1753 (2012)Google Scholar
  14. 14.
    A. Deya, I. Nourdin, Convergence of Wigner integrals to the tetilla law. ALEA 9, 101–127 (2012)MathSciNetGoogle Scholar
  15. 15.
    C.G. Esseen, A moment inequality with an application to the central limit theorem. Skand. Aktuarietidskr. 39, 160–170 (1956)MathSciNetGoogle Scholar
  16. 16.
    S.-T. Ho, L.H.Y. Chen, An L p bound for the remainder in a combinatorial central limit theorem. Ann. Probab. 6(2), 231–249 (1978)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    T. Kemp, I. Nourdin, G. Peccati, R. Speicher, Wigner chaos and the fourth moment. Ann. Probab. 40(4), 1577–1635 (2012)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    V.Yu. Korolev, I.G. Shevtsova, On the upper bound for the absolute constant in the Berry-Esseen inequality. Theory Probab. Appl. 54(4), 638–658 (2010)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    R. Lachièze-Rey, G. Peccati, Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs. arXiv preprint arXiv:1111.7312 (2011)Google Scholar
  20. 20.
    E. Mossel, R. O’Donnell, K. Oleszkiewicz, Noise stability of functions with low influences: invariance and optimality. Ann. Math. 171(1), 295–341 (2010)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    E. Nelson, The free Markoff field. J. Funct. Anal. 12, 211–227 (1973)MATHCrossRefGoogle Scholar
  22. 22.
    A. Nica, R. Speicher, Lectures on the Combinatorics of Free Probability (Cambridge University Press, Cambridge, 2006)MATHCrossRefGoogle Scholar
  23. 23.
    S. Noreddine, I. Nourdin, On the Gaussian approximation of vector-valued multiple integrals. J. Multiv. Anal. 102(6), 1008–1017 (2011)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    I. Nourdin, Yet another proof of the Nualart-Peccati criterion. Electron. Comm. Probab. 16, 467–481 (2011)MathSciNetMATHGoogle Scholar
  25. 25.
    I. Nourdin, Selected Aspects of Fractional Brownian Motion (Springer, New York, 2012)MATHCrossRefGoogle Scholar
  26. 26.
    I. Nourdin, G. Peccati, Non-central convergence of multiple integrals. Ann. Probab. 37(4), 1412–1426 (2007)MathSciNetCrossRefGoogle Scholar
  27. 27.
    I. Nourdin, G. Peccati, Stein’s method on Wiener chaos. Probab. Theory Rel. Fields 145(1), 75–118 (2009)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    I. Nourdin, G. Peccati, Stein’s method meets Malliavin calculus: a short survey with new estimates. Recent Advances in Stochastic Dynamics and Stochastic Analysis (World Scientific, Singapore, 2010), pp. 207–236Google Scholar
  29. 29.
    I. Nourdin, G. Peccati, Cumulants on the Wiener space. J. Funct. Anal. 258, 3775–3791 (2010)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    I. Nourdin, G. Peccati, Stein’s method and exact Berry-Esseen asymptotics for functionals of Gaussian fields. Ann. Probab. 37(6), 2231–2261 (2010)MathSciNetCrossRefGoogle Scholar
  31. 31.
    I. Nourdin, G. Peccati, Poisson approximations on the free Wigner chaos. arXiv preprint arXiv:1103.3925 (2011)Google Scholar
  32. 32.
    I. Nourdin, G. Peccati, Normal Approximations Using Malliavin Calculus: from Stein’s Method to Universality. Cambridge Tracts in Mathematics (Cambridge University Press, Cambridge, 2012)Google Scholar
  33. 33.
    I. Nourdin, G. Peccati, M. Podolskij, Quantitative Breuer–Major Theorems. Stoch. Proc. App. 121(4), 793–812 (2011)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    I. Nourdin, G. Peccati, G. Reinert, Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos. Ann. Probab. 38(5), 1947–1985 (2010)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    I. Nourdin, G. Peccati, A. Réveillac, Multivariate normal approximation using Stein’s method and Malliavin calculus. Ann. Inst. H. Poincaré (B) Probab. Stat. 46(1), 45–58 (2010)Google Scholar
  36. 36.
    I. Nourdin, G. Peccati, R. Speicher, Multidimensional semicircular limits on the free Wigner chaos, in Ascona Proceedings, Birkhäuser Verlag (2013)Google Scholar
  37. 37.
    I. Nourdin, F.G. Viens, Density estimates and concentration inequalities with Malliavin calculus. Electron. J. Probab. 14, 2287–2309 (2009) (electronic)Google Scholar
  38. 38.
    D. Nualart, The Malliavin Calculus and Related Topics, 2nd edn. (Springer, Berlin, 2006)MATHGoogle Scholar
  39. 39.
    D. Nualart, S. Ortiz-Latorre, Central limit theorems for multiple stochastic integrals and Malliavin calculus. Stoch. Proc. Appl. 118(4), 614–628 (2008)MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    D. Nualart, G. Peccati, Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33(1), 177–193 (2005)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    D. Nualart, L. Quer-Sardanyons, Optimal Gaussian density estimates for a class of stochastic equations with additive noise. Infinite Dimensional Anal. Quant. Probab. Relat. Top. 14(1), 25–34 (2011)MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    D. Nualart, J. Vives, Anticipative calculus for the Poisson process based on the Fock space. Séminaire de Probabilités, vol. XXIV, LNM 1426 (Springer, New York, 1990), pp. 154–165Google Scholar
  43. 43.
    G. Peccati, The Chen-Stein method for Poisson functionals. arXiv:1112.5051v3 (2012)Google Scholar
  44. 44.
    G. Peccati, J.-L. Solé, M.S. Taqqu, F. Utzet, Stein’s method and normal approximation of Poisson functionals. Ann. Probab. 38(2), 443–478 (2010)MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    G. Peccati, M.S. Taqqu, Wiener Chaos: Moments, Cumulants and Diagrams (Springer, New York, 2010)Google Scholar
  46. 46.
    G. Peccati, C.A. Tudor, Gaussian limits for vector-valued multiple stochastic integrals. Séminaire de Probabilités, vol. XXXVIII, LNM 1857 (Springer, New York, 2005), pp. 247–262Google Scholar
  47. 47.
    G. Peccati, C. Zheng, Multidimensional Gaussian fluctuations on the Poisson space. Electron. J. Probab. 15, paper 48, 1487–1527 (2010) (electronic)Google Scholar
  48. 48.
    G. Peccati, C. Zheng, Universal Gaussian fluctuations on the discrete Poisson chaos. arXiv preprint arXiv:1110.5723v1Google Scholar
  49. 49.
    M. Penrose, Random Geometric Graphs (Oxford University Press, Oxford, 2003)MATHCrossRefGoogle Scholar
  50. 50.
    W. Rudin, Real and Complex Analysis, 3rd edn. (McGraw-Hill, New York, 1987)MATHGoogle Scholar
  51. 51.
    I. Shigekawa, Derivatives of Wiener functionals and absolute continuity of induced measures. J. Math. Kyoto Univ. 20(2), 263–289 (1980)MathSciNetMATHGoogle Scholar
  52. 52.
    Ch. Stein, A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. II: Probability Theory, 583–602. University of California Press, Berkeley, California (1972)Google Scholar
  53. 53.
    D.W. Stroock, Homogeneous chaos revisited. In: Séminaire de Probabilités, vol. XXI. Lecture Notes in Math. vol. 1247 (Springer, Berlin, 1987), pp. 1–8Google Scholar
  54. 54.
    M. Talagrand, Spin Glasses, a Challenge for Mathematicians (Springer, New York, 2003)MATHGoogle Scholar
  55. 55.
    D.V. Voiculescu, Symmetries of some reduced free product C  ∗ -algebras. Operator algebras and their connection with topology and ergodic theory, Springer Lecture Notes in Mathematics, vol. 1132, 556–588 (1985)Google Scholar
  56. 56.
    R. Zintout, Total variation distance between two double Wiener-Itô integrals. Statist. Probab. Letter, to appear (2013)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.Institut de Mathématiques Élie CartanUniversité de LorraineVandoeuvre-lès-Nancy CedexFrance

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