Top

Published in:

2013 | OriginalPaper | Chapter

Lectures on Gaussian Approximations with Malliavin Calculus

Author : Ivan Nourdin

Published in: Séminaire de Probabilités XLV

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

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.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

next chapter Some Sufficient Conditions for the Ergodicity of the Lévy Transformation

You may watch the videos of the lectures at http://www.sciencesmaths-paris.fr/index.php?page=175.

Any adapted process u that is either càdlàg or càglàd admits a progressively measurable version. We will always assume that we are dealing with it.

B. Bercu, I. Nourdin, M.S. Taqqu, Almost sure central limit theorems on the Wiener space. Stoch. Proc. Appl. 120(9), 1607–1628 (2010)MathSciNetMATHCrossRef

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)MathSciNetMATHCrossRef

H. Biermé, A. Bonami, J. Léon, Central limit theorems and quadratic variations in terms of spectral density. Electron. J. Probab. 16, 362–395 (2011)MathSciNetMATHCrossRef

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)

E. Bolthausen, An estimate of the remainder in a combinatorial central limit theorem. Z. Wahrscheinlichkeitstheorie verw. Gebiete 66, 379–386 (1984)MathSciNetMATHCrossRef

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)

P. Breuer, P. Major, Central limit theorems for non-linear functionals of Gaussian fields. J. Mult. Anal. 13, 425–441 (1983)MathSciNetMATHCrossRef

L.H.Y. Chen, Poisson approximation for dependent trials. Ann. Probab. 3(3), 534–545 (1975)MATHCrossRef

L.H.Y. Chen, L. Goldstein, Q.-M. Shao, Normal Approximation by Stein’s Method. Probability and Its Applications. Springer, New York (2010)

10.

F. Daly, Upper bounds for Stein-type operators. Electron. J. Probab. 13(20), 566–587 (2008) (electronic)

11.

L. Decreusefond, D. Nualart, Hitting times for Gaussian processes. Ann. Probab. 36(1), 319–330 (2008)MathSciNetMATHCrossRef

12.

A. Deya, S. Noreddine, I. Nourdin, Fourth moment theorem and q-Brownian chaos. Commun. Math. Phys. 1–22 (2012)

13.

A. Deya, I. Nourdin, Invariance principles for homogeneous sums of free random variables. arXiv preprint arXiv:1201.1753 (2012)

14.

A. Deya, I. Nourdin, Convergence of Wigner integrals to the tetilla law. ALEA 9, 101–127 (2012)MathSciNet

15.

C.G. Esseen, A moment inequality with an application to the central limit theorem. Skand. Aktuarietidskr. 39, 160–170 (1956)MathSciNet

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)MathSciNetMATHCrossRef

17.

T. Kemp, I. Nourdin, G. Peccati, R. Speicher, Wigner chaos and the fourth moment. Ann. Probab. 40(4), 1577–1635 (2012)MathSciNetMATHCrossRef

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)MathSciNetMATHCrossRef

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)

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)MathSciNetMATHCrossRef

21.

E. Nelson, The free Markoff field. J. Funct. Anal. 12, 211–227 (1973)MATHCrossRef

22.

A. Nica, R. Speicher, Lectures on the Combinatorics of Free Probability (Cambridge University Press, Cambridge, 2006)MATHCrossRef

23.

S. Noreddine, I. Nourdin, On the Gaussian approximation of vector-valued multiple integrals. J. Multiv. Anal. 102(6), 1008–1017 (2011)MathSciNetMATHCrossRef

24.

I. Nourdin, Yet another proof of the Nualart-Peccati criterion. Electron. Comm. Probab. 16, 467–481 (2011)MathSciNetMATH

25.

I. Nourdin, Selected Aspects of Fractional Brownian Motion (Springer, New York, 2012)MATHCrossRef

26.

I. Nourdin, G. Peccati, Non-central convergence of multiple integrals. Ann. Probab. 37(4), 1412–1426 (2007)MathSciNetCrossRef

27.

I. Nourdin, G. Peccati, Stein’s method on Wiener chaos. Probab. Theory Rel. Fields 145(1), 75–118 (2009)MathSciNetMATHCrossRef

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. 207–236

29.

I. Nourdin, G. Peccati, Cumulants on the Wiener space. J. Funct. Anal. 258, 3775–3791 (2010)MathSciNetMATHCrossRef

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)MathSciNetCrossRef

31.

I. Nourdin, G. Peccati, Poisson approximations on the free Wigner chaos. arXiv preprint arXiv:1103.3925 (2011)

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)

33.

I. Nourdin, G. Peccati, M. Podolskij, Quantitative Breuer–Major Theorems. Stoch. Proc. App. 121(4), 793–812 (2011)MathSciNetMATHCrossRef

34.

I. Nourdin, G. Peccati, G. Reinert, Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos. Ann. Probab. 38(5), 1947–1985 (2010)MathSciNetMATHCrossRef

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)

36.

I. Nourdin, G. Peccati, R. Speicher, Multidimensional semicircular limits on the free Wigner chaos, in Ascona Proceedings, Birkhäuser Verlag (2013)

37.

I. Nourdin, F.G. Viens, Density estimates and concentration inequalities with Malliavin calculus. Electron. J. Probab. 14, 2287–2309 (2009) (electronic)

38.

D. Nualart, The Malliavin Calculus and Related Topics, 2nd edn. (Springer, Berlin, 2006)MATH

39.

D. Nualart, S. Ortiz-Latorre, Central limit theorems for multiple stochastic integrals and Malliavin calculus. Stoch. Proc. Appl. 118(4), 614–628 (2008)MathSciNetMATHCrossRef

40.

D. Nualart, G. Peccati, Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33(1), 177–193 (2005)MathSciNetMATHCrossRef

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)MathSciNetMATHCrossRef

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. 154–165

43.

G. Peccati, The Chen-Stein method for Poisson functionals. arXiv:1112.5051v3 (2012)

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)MathSciNetMATHCrossRef

45.

G. Peccati, M.S. Taqqu, Wiener Chaos: Moments, Cumulants and Diagrams (Springer, New York, 2010)

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. 247–262

47.

G. Peccati, C. Zheng, Multidimensional Gaussian fluctuations on the Poisson space. Electron. J. Probab. 15, paper 48, 1487–1527 (2010) (electronic)

48.

G. Peccati, C. Zheng, Universal Gaussian fluctuations on the discrete Poisson chaos. arXiv preprint arXiv:1110.5723v1

49.

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

50.

W. Rudin, Real and Complex Analysis, 3rd edn. (McGraw-Hill, New York, 1987)MATH

51.

I. Shigekawa, Derivatives of Wiener functionals and absolute continuity of induced measures. J. Math. Kyoto Univ. 20(2), 263–289 (1980)MathSciNetMATH

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)

53.

D.W. Stroock, Homogeneous chaos revisited. In: Séminaire de Probabilités, vol. XXI. Lecture Notes in Math. vol. 1247 (Springer, Berlin, 1987), pp. 1–8

54.

M. Talagrand, Spin Glasses, a Challenge for Mathematicians (Springer, New York, 2003)MATH

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)

56.

R. Zintout, Total variation distance between two double Wiener-Itô integrals. Statist. Probab. Letter, to appear (2013)

Title: Lectures on Gaussian Approximations with Malliavin Calculus
Author: Ivan Nourdin
Publisher: Springer International Publishing
Book: Séminaire de Probabilités XLV
Print ISBN: 978-3-319-00320-7

Electronic ISBN: 978-3-319-00321-4

Copyright Year: 2013
DOI: https://doi.org/10.1007/978-3-319-00321-4_1