Why this webpage?

  In a seminal paper of 2005, Nualart and Peccati discovered a surprising central limit theorem (called the ``fourth moment theorem'' in the sequel; alternative proofs can be found here, here and here) for sequences of multiple stochastic integrals of a fixed order: in this context, convergence in distribution to the standard normal law is actually equivalent to convergence of just the fourth moment! Shortly afterwards, Peccati and Tudor gave a multidimensional version of this characterization.
  Since the publication of these two beautiful papers, many improvements and developments on this theme have been considered. Among them is the work by Nualart and Ortiz-Latorre, giving a new proof only based on Malliavin calculus and the use of integration by parts on Wiener space. A second step is my joint paper ``Stein's method on Wiener chaos" written in collaboration with Peccati in which, by bringing together Stein's method with Malliavin calculus, we have been able (among other things) to associate quantitative bounds to the fourth moment theorem.
  It turns out that Stein's method and Malliavin calculus fit together admirably well, and that their interaction has led to some remarkable new results involving central and non-central limit theorems for functionals of infinite-dimensional Gaussian fields.
  This webpage aims to gather all the available ressources (including research papers, slides of talks, lecture notes, etc.) having any link with the fourth moment theorem and related stuff. Although my goal has been to be as comprehensive as possible, several links are surely missing, so please feel free to contact me (inourdin@gmail.com) in case.

Expository papers or (part of) books

- V. Bogachev (2010): Differentiable Measures and the Malliavin Calculus, American Mathematical Society (see more precisely pages 321-323)

- L. H. Y. Chen, L. Goldstein and Q.-M. Shao (2011): Normal Approximation by Stein’s Method, Probability and Its Applications, Springer-Verlag (see more precisely the chapter 14, entitled ``Group Characters and Malliavin Calculus'')

- J.M. Corcuera (2011): New Central Limit Theorems for Functionals of Gaussian Processes and their Applications. Methodol. Comput. Appl. Probab. (online first)

- D. Marinucci and G. Peccati (2011): Random fields on the Sphere. Representation, Limit Theorems and Cosmological Applications. Series: London Mathematical Society Lecture Note Series 389. Cambridge University Press.

- I. Nourdin and G. Peccati (2010): Stein's method meets Malliavin calculus: a short survey with new estimates, In the volume: Recent Advances in Stochastic Dynamics and Stochastic Analysis, World Scientific

- I. Nourdin and G. Peccati (2011): Normal approximations with Malliavin calculus: from Stein's method to universality. Forthcoming book

- D. Nualart (2009): Malliavin Calculus and Its Applications, American Mathematical Society and CBMS Regional Conference Series in Mathematics (see more precisely the chapter 9, entitled ``Central limit theorem and Malliavin calculus'')

- G. Peccati and M.S. Taqqu (2010): Wiener Chaos: Moments, Cumulants and Diagrams, Springer-Verlag (see more precisely the chapter 11, entitled ``Limit theorems on the Gaussian Wiener chaos'')

- G. Reinert (2009): Gaussian approximation of functionals: Malliavin calculus and Stein’s method, Proceedings of the 33rd Conference on Stochastic Processes and Their Applications, to appear

Lecture notes

- S. Chatterjee (2007): Stein's method and applications, University of Berkeley

- G. Peccati (2009): Stein's method, Malliavin calculus and infinite-dimensional Gaussian analysis, Progress in Stein’s Method, Singapore

- I. Nourdin (2012): Lectures on Gaussian approximations with Malliavin calculus, Prix de la Fondation des Sciences Mathématiques de Paris

Talks

- A. Bonami (2011): Quantitative Central Limit Theorems, Conference ``Analysis 2011, in honor of Martine and Hervé Queffélec'', June 14-17, University of Lille 1

- F. Lu (2011): Malliavin Calculus, Its Applications to SPDE and Convergence in Density, oral comprehensive exam, October 19, University of Kansas

- I. Nourdin (2010): Convergence des intégrales multiples de Wigner vers la loi semi-circulaire, 50th anniversary of the Laboratoire de Probabilités et Modèles Aléatoires, December 6-7, University of Paris 6

- I. Nourdin (2012): Approximation gaussienne à l’aide du calcul de Malliavin: de la méthode de Stein à l’universalité, Cours de la Fondation des Sciences Mathématiques de Paris, January to March, Collège de France

- S. Ortiz-Latorre (2009): An introduction to Stein’s Method, Barcelona Probability Seminar

- G. Peccati (2010): Universal Gaussian fluctuations on the Wiener chaos, Conference on Probability and Stochastic Processes, August 13-17, Indian Statistical Institute, Bangalore

- G. Peccati (2010): Stein's method and Malliavin calculus on the Poisson space, Workshop on Malliavin Calculus for Jump Processes, November 18-20, Université Paris-Est

- M. Podolskij (2010): Asymptotic expansion for multiple integrals, International Conference Dynstoch Meeting, June 16-19, University of Angers

- G. Reinert (2009): Stein’s method and stochastic analysis of Rademacher functionals, Progress in Stein’s Method, January 12-16, National University of Singapore

- G. Reinert (2009): Gaussian approximation using Stein's method: from one to infinite dimensions and back, Symposium on probability theory in honor of Professor Andrew Barbour, December 4th, Universität Zürich

- A. Rohde and C. Strauch (2010): Malliavin-Kalkül und Steinsche Methode, Universität Hamburg

- C.A. Tudor (2010): Hsu-Robbins theorem for the correlated sequences, Colloque Franco-Roumain de Mathématiques Appliquées, August 2010, Poitiers, France.

- F.G. Viens (2010): Application of Stein's lemma and Malliavin calculus to the densities and fluctuation exponents of stochastic heat equations, Conference on SPDEs, January 6th, Isaac Newton Institute for Mathematical Sciences

Research articles whose aim is to improve the theory

- H. Airault, P. Malliavin and F.G. Viens (2010): Stokes formula on the Wiener space and n-dimensional Nourdin–Peccati analysis, J. Funct. Anal. 258, 1763-1783

- B. Bercu, I. Nourdin and M.S. Taqqu (2010): Almost sure central limit theorems on the Wiener space , Stoch. Proc. Appl. 120, no. 9, 1607-1628

- H. Biermé, A. Bonami and J.R. León (2011): Central limit theorems and quadratic variations in terms of spectral density, Electron. J. Probab. 16, no. 13, 362-395

- H. Biermé, A. Bonami, I. Nourdin and G. Peccati (2011): Optimal Berry-Esseen rates on the Wiener space: the barrier of third and fourth cumulants, preprint

- S. Bourguin and C.A. Tudor (2011): Cramér's theorem for Gamma random variables, Electron. Comm. Probab., to appear

- A. Deya and I. Nourdin (2011): Convergence of Wigner integrals to the tetilla law, ALEA, to appear

- A. Deya and I. Nourdin (2011): Invariance principles for homogeneous sums of free random variables, preprint

- A. Deya, S. Noreddine and I. Nourdin (2012): Fourth Moment Theorem and q-Brownian Chaos, preprint

- R. Eden and F.G. Viens (2010): General upper and lower tail estimates using Malliavin calculus and Stein's equations, Preprint

- R. Eden and J. Víquez (2012): Nourdin-Peccati analysis on Wiener and Wiener-Poisson space for general distributions, Preprint

- T. Kemp, I. Nourdin, G. Peccati and R. Speicher (2010): Wigner chaos and the fourth moment, Ann. Probab., to appear

- S. Kusuoka and C.A. Tudor (2011): Stein's method for invariant measures of diffusions via Malliavin calculus, Preprint

- R. Lachieze-Rey and G. Peccati (2011): Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs, preprint

- R. Lachieze-Rey and G. Peccati (2012): Fine Gaussian fluctuations on the Poisson space II: rescaled kernels, marked processes and geometric U-statistics, preprint

- M. Ledoux (2010): Chaos of a Markov operator and the fourth moment condition, Ann. Probab., to appear

- S. Noreddine and I. Nourdin (2011): On the Gaussian approximation of vector-valued multiple integrals, J. Multiv. Anal. 102, no. 6, 1008-1017.

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

- I. Nourdin and D. Nualart (2010): Central limit theorems for multiple Skorohod integrals, J. Theoret. Probab. 23, no. 1, 39-64

- I. Nourdin and G. Peccati (2009): Non-central convergence of multiple integrals, Ann. Probab. 37, no. 4, 1412–1426

- I. Nourdin and G. Peccati (2009): Stein's method on Wiener chaos, Probab. Theory Rel. Fields 145, no. 1, 75-118

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

- I. Nourdin and G. Peccati (2010): Stein's method and exact Berry-Esséen asymptotics for functionals of Gaussian fields, Ann. Probab. 37, no. 6, 2231-2261

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

- I. Nourdin, G. Peccati and G. Reinert (2009): Second order Poincaré inequalities and CLTs on Wiener space, J. Funct. Anal. 257, 593-609

- I. Nourdin, G. Peccati and G. Reinert (2010): Stein's method and stochastic analysis of Rademacher sequences, Elect. J. Probab. 15, no. 55, 1703-1742

- I. Nourdin, G. Peccati and G. Reinert (2010): Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos, Ann. Probab. 38, no. 5, 1947-1985

- I. Nourdin, G. Peccati and A. Réveillac (2010): Multivariate normal approximation using Stein's method and Malliavin calculus, Ann. I.H.P. 46, no. 1, 45-58

- I. Nourdin, G. Peccati and R. Speicher (2011): Multidimensional semicircular limits on the free Wigner chaos, preprint

- I. Nourdin and G. Poly (2012): Convergence in law in the second Wiener/Wigner chaos, Preprint

- I. Nourdin and G. Poly (2012): Convergence in total variation on Wiener chaos, Preprint

- I. Nourdin and J. Rosinski (2011): Asymptotic independence of multiple Wiener-Itô integrals and the resulting limit laws, Preprint

- I. Nourdin and F.G. Viens (2009): Density formula and concentration inequalities with Malliavin calculus, Electron. J. Probab. 14, 2287-2309

- D. Nualart and S. Ortiz-Latorre (2008): Central limit theorems for multiple stochastic integrals and Malliavin calculus, Stoch. Proc. Appl. 118, no. 4, 614-628

- D. Nualart and G. Peccati (2005): Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. 33, no. 1, 177-193

- G. Peccati (2007): Gaussian approximations of multiple integrals, Elect. Comm. in Probab. 12, 350–364

- G. Peccati (2011): The Chen-Stein method for Poisson functionals, Preprint

- G. Peccati, J.-L. Solé, M.S. Taqqu and F. Utzet (2010): Stein's method and normal approximation of Poisson functionals, Ann. Probab. 38, no. 2, 443-478

- G. Peccati and C.A. Tudor (2004): Gaussian limits for vector-valued multiple stochastic integrals, Séminaire de Probabilités XXXVIII, 247-262

- G. Peccati and C. Zheng (2010): Multi-dimensional Gaussian fluctuations on the Poisson space, Elect. J. Probab. 15, no. 48, 1487-1527

- G. Peccati and C. Zheng (2011): Universal Gaussian fluctuations on the discrete Poisson chaos, Preprint

- C.A. Tudor (2011): Asymptotic Cramér's theorem and analysis on Wiener space, Séminaire de Probabilités XLIII, Lecture Notes in Mathematics, 309-325

- F.G. Viens (2009): Stein's lemma, Malliavin calculus, and tail bounds, with application to polymer fluctuation exponent, Stoch. Proc. Appl. 119, 3671-3698

- J. Víquez (2011): On the second order Poincaré inequality and CLT on Wiener-Poisson space, Preprint

Research articles applying the theory

- S. Aazizi and K. Es-Sebaiy (2012): Berry-Esseen bounds and almost sure CLT for the quadratic variation of the bifractional Brownian motion. Preprint

O. Aboura and S. Bourguin (2011): Density estimates for solutions to one dimensional SDE's and Backward SDE's. Preprint

- F. Avram, N. Leonenko and L. Sakhno (2012): Limit theorems for additive functionals of stationary fields, under integrability assumptions on the higher order spectral densities. Preprint

- P. Baldi, G. Kerkyacharian, D. Marinucci and D. Picard (2009): Asymptotics for spherical needlets, Ann. Statist. 37, no. 3, 1150-1171.

- O. E. Barndorff-Nielsen, J. M. Corcuera and M. Podolskij (2009): Power variation for Gaussian processes with stationary increments, Stoch. Proc. Appl. 119, 1845-1865

- O. E. Barndorff-Nielsen, J. M. Corcuera and M. Podolskij (2010): Multipower variation for Brownian semi-stationary processes, Bernoulli, to appear

- O. E. Barndorff-Nielsen, J. M. Corcuera, M. Podolskij and J. H. C. Woerner (2009): Bipower variation for Gaussian processes with stationary increments, J. Appl. Probab. 46, 132-150

- S. Bourguin and C.A. Tudor (2010): Berry-Esseen bounds for long memory moving averages via Stein's method and Malliavin calculus. Stoch. Anal. Appl., to appear

- S. Bourguin and C.A. Tudor (2011): Malliavin Calculus and Self Normalized Sums, Preprint

- J.-C. Breton and J.-F. Coeurjolly (2009): Confidence intervals for the Hurst parameter of a fractional Brownian motion based on finite sample size, Preprint

- J.-C. Breton and I. Nourdin (2008): Error bounds on the non-normal approximation of Hermite power variations of fractional Brownian motion, Electron. Comm. in Probab. 13, 482-493

- J.-C. Breton, I. Nourdin and G. Peccati (2009): Exact confidence intervals for the Hurst parameter of a fractional Brownian motion, Electron. J. Statist. 3, 416-425

- B. Buchmann and N. H. Chan (2009): Integrated functionals of normal and fractional processes, Ann. Appl. Probab. 19, no. 1, 49-70.

- J.M. Corcuera, D. Nualart and J.H.C. Woerner (2006): Power variation of some integral fractional processes, Bernoulli 12, no. 4, 713-735

- S. Darses, I. Nourdin and D. Nualart (2010): Limit theorems for nonlinear functionals of Volterra processes via white noise analysis, Bernoulli 16, no. 4, 1262-1293

- L. Decreusefond, E. Ferraz and H. Randriam (2011): Simplicial Homology of Random Configurations, Preprint

- L. Decreusefond, E. Ferraz, P. Martins and T. Vu (2012): Robust methods for LTE and WiMAX dimensioning, Preprint

- C. Durastanti, X. Lan and D. Marinucci (2011): Gaussian Semiparametric Estimates on the Unit Sphere. Preprint.

- C. Durastanti, X. Lan and D. Marinucci (2012): Needlet-Whittle Estimates on the Unit Sphere. Preprint.

- K. Es-Sebaiy (2012): Berry-Esseen bounds for the least squares estimator for discretely observed fractional Ornstein-Uhlenbeck processes , Preprint

- K. Es-Sebaiy and C. Tudor (2011): Noncentral limit theorem for the cubic variation of a class of self-similar stochastic processes, Theory Probab. Appl. 55, no. 3, 411-431.

- E. Ferraz and A. Vergne (2011): Statistics of geometric random simplicial complexes, Preprint

- Y. Hu and D. Nualart (2005): Renormalized self-intersection local time for fractional Brownian motion, Ann. Probab. 33, no. 3, 948-983

- Y. Hu and D. Nualart (2010): Parameter estimation for fractional Ornstein-Uhlenbeck processes, Stat. Probab. Lett. 80, no. 11-12, 1030-1038

- Y. Hu, D. Nualart, X. Weilin and Z. Weiguo (2011): Exact maximum likelihood estimator for drift fractional Brownian motion at discrete observation, Acta Math. Scientia 31B, no. 5, 1851-1859

- X. Lan and D. Marinucci (2008): The needlets bispectrum, Electron. J. Statist. 2, 332-367

- X. Lan and D. Marinucci (2009): On The Dependence Structure of Wavelet Coefficients for Spherical Random Fields, Stoch. Proc. Appl. 119 3749-3766

- G. Last, M. D. Penrose, M. Schulte and C. Thaele (2012): Moments and central limit theorems for some multivariate Poisson functionals, preprint

- D. Marinucci and G. Peccati (2008): High-frequency asymptotics for subordinated isotropic fields on an Abelian compact group, Stoch. Proc. Appl. 118, no. 4, 585-613

- D. Marinucci and G. Peccati (2010): Group representations and high-resolution central limit theorems for subordinated spherical random fields. Bernoulli 16, no. 3, 798-824.

- D. Marinucci and G. Peccati (2010): Ergodicity and Gaussianity for Spherical Random Fields, J. Math. Phys. 51, 043301

- D. Marinucci and I. Wigman (2010): On the Excursion Sets of Spherical Gaussian Eigenfunctions, preprint

- A. Neuenkirch and I. Nourdin (2007): Exact rate of convergence of some approximation schemes associated to SDEs driven by a fBm. J. Theoret. Probab. 20, 871-899

- A. Neuenkirch, S. Tindel and J. Unterberger (2010): Discretizing the fractional Levy area, Stoch. Proc. Appl. 120, no. 2, 223-254

- M.T. Nguyen (2011): Malliavin-Stein method for multi-dimensional U-statistics of Poisson point processes, preprint

- I. Nourdin (2009): A change of variable formula for the 2D fractional Brownian motion of Hurst index bigger or equal to 1/4, J. Funct. Anal. 256, 2303-2320

- I. Nourdin, D. Nualart and C.A. Tudor (2010): Central and non-central limit theorems for weighted power variations of fractional Brownian motion, Ann. I.H.P. 46, no. 4, 1055-1079

- I. Nourdin and G. Peccati (2008): Weighted power variations of iterated Brownian motion, Elect. J. Probab. 13, no. 43, 1229-1256

- I. Nourdin and G. Peccati (2010): Universal Gaussian fluctuations of non-Hermitian matrix ensembles: from weak convergence to almost sure CLTs, ALEA 7, 341-375

- I. Nourdin, G. Peccati and M. Podolskij (2011): Quantitative Breuer-Major theorems. Stoch. Proc. Appl. 121, no. 4, 793-812.

- I. Nourdin and A. Réveillac (2009): Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: the critical case H=1/4, Ann. Probab. 37, no. 6, 2200-2230

- I. Nourdin, A. Réveillac and J. Swanson (2010): The weak Stratonovich integral with respect to fractional Brownian motion with Hurst parameter 1/6. Elect. J. Probab. 15, 2117-2162.

- I. Nourdin and M.S. Taqqu (2011): Central and non-central limit theorems in a free probability setting. Preprint.

- D. Nualart and L. Quer-Sardanyons(2009): Gaussian density estimates for solutions to quasi-linear stochastic partial differential equations. Stoch. Proc. Appl. 119, 3914-3938.

- D. Nualart and L. Quer-Sardanyons(2009): Optimal Gaussian density estimates for a class of stochastic equations with additive noise. Infinite Dimensional Analysis, Quantum Probability and Related Topics, to appear.

- H.S. Park, J.W. Jeon and Y.T. Kim (2010): The central limit theorem for cross-variation related to the standard Brownian sheet and Berry–Esseen bounds, J. Korean Statist. Soc., in press

- M. Reitzner and M. Schulte (2011): Central Limit Theorems for U-Statistics of Poisson Point Processes, Preprint

- A. Réveillac (2009): Convergence of finite-dimensional laws of the weighted quadratic variations process for some fractional Brownian sheets, Stoch. Anal. Appl. 27, no. 1, 51-73

- A. Réveillac, M. Stauch and C.A. Tudor (2010): Hermite variations of the fractional Brownian sheet. Stoch. Dyn., to appear.

- M. Schulte (2011): A Central Limit Theorem for the Poisson-Voronoi Approximation, Preprint

- M. Schulte and C. Thaele (2010): Exact and asymptotic results for intrinsic volumes of Poisson k-flat processes, Preprint

- M. Schulte and C. Thaele (2012): The scaling limit of Poisson-driven order statistics with applications in geometric probability, preprint

- S. Si (2009): Two-step variations for processes driven by fractional Brownian motion with application in testing for jumps from the high frequency data, PhD thesis, University of Tennessee

- C.A. Tudor (2009): Hsu-Robbins and Spitzer's theorems for the variations of fractional Brownian motion, Elect. Comm. in Probab. 14, 278–289

- C. Tudor (2011): Berry–Esséen bounds and almost sure CLT for the quadratic variation of the sub-fractional Brownian motion, J. Math. Anal. Appl. 375, no. 2, 667-676.