malliavinstein

A webpage maintained by Ivan Nourdin

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 pathbreaking 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 were 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 resources research papers having any link with the fourth moment theorem and related stuff. I have tried to be as comprehensive as possible, but several links are surely missing. In case, please feel free to contact me (inourdin@gmail.com).

1. S. Bourguin, C. Durastanti, D. Marinucci and G. Peccati (2014): Gaussian approximations of nonlinear statistics on the sphere

O. Aboura and S. Bourguin: Density estimates for solutions to one dimensional backward SDE's. Potential Analysis 38, no. 2 (2013), pp 573-587
O. E. Barndorff-Nielsen, J. M. Corcuera and M. Podolskij (2011): Multipower variation for Brownian semi-stationary processes, Bernoulli 17, no. 4, 1159-1194.
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
S. Bourguin and C.A. Tudor: Malliavin Calculus and Self Normalized Sums, Séminaire de Probabilités XLV, Lecture Notes in Mathematics 2013, pp 323-351
S. Bourguin and C.A. Tudor (2011): Berry-Esseen bounds for long memory moving averages via Stein's method and Malliavin calculus. Stoch. Anal. Appl. 29, no. 5, 881-905
S. Bourguin and C.A. Tudor (2011): Cramér's theorem for Gamma random variables, Electron. Comm. Probab. 16, no. 1, 365-378.
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)
L. Decreusefond, E. Ferraz and H. Randriam: Simplicial Homology of Random Configurations, Journal of Advances in Applied Probability, Mars 2013
A. Deya and I. Nourdin: Invariance principles for homogeneous sums of free random variables, Bernoulli 20, no. 2 (2014), 586-603
C. Durastanti, X. Lan and D. Marinucci: Gaussian Semiparametric Estimates on the Unit Sphere, Bernoulli 20, no. 1 (2014), 28-77
K. Es-Sebaiy and C.A. 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
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
R. Lachieze-Rey and G. Peccati: Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs, Electron. J. Probab.18 (2013), no. 32, 1–32.
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.
M.T. Nguyen (2011): Malliavin-Stein method for multi-dimensional U-statistics of Poisson point processes
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
1. I. Nourdin and G. Peccati: Poisson approximations on the free Wigner chaos, Ann. Probab. 41, no. 4 (2013), 2709-2723
I. Nourdin, G. Peccati and M. Podolskij (2011): Quantitative Breuer-Major theorems. Stoch. Proc. Appl. 121, no. 4, 793-812.
I. Nourdin, G. Peccati and R. Speicher: Multidimensional semicircular limits on the free Wigner chaos, Seminar on Stochastic Analysis, Random Fields and Applications VII, Progress in Probability 67, 2013, pp 211-221
I. Nourdin and J. Rosiński: Asymptotic independence of multiple Wiener-Itô integrals and the resulting limit laws, Ann. Probab. 42, no. 2 (2014), 497-526
I. Nourdin and M.S. Taqqu (2011): Central and non-central limit theorems in a free probability setting, J. Theoret. Probab. 27, no. 1, 220-248
D. Nualart and L. Quer-Sardanyons (2011): Optimal Gaussian density estimates for a class of stochastic equations with additive noise. Infinite Dimensional Analysis, Quantum Probability and Related Topics 14, 25-34.
H.S. Park, J.W. Jeon and Y.T. Kim (2011): The central limit theorem for cross-variation related to the standard Brownian sheet and Berry–Esseen bounds, J. Korean Statist. Soc. 40, no. 2, 239-244
G. Peccati (2011): The Chen-Stein method for Poisson functionals
G. Peccati and C. Zheng: Universal Gaussian fluctuations on the discrete Poisson chaos, Bernoulli 20, no. 2 (2014), 697-715

1. M. Reitzner and M. Schulte: Central Limit Theorems for U-Statistics of Poisson Point Processes, Ann. Probab. 41, no. 6 (2013), 3879-3909

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
V. Bogachev (2010): Differentiable Measures and the Malliavin Calculus, American Mathematical Society (see more precisely pages 321-323)
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
R. Eden and F.G. Viens: General upper and lower tail estimates using Malliavin calculus and Stein's equations, Seminar on Stochastic Analysis, Random Fields and Applications VII Progress in Probability Volume 67, 2013, pp 55-84
Y. Hu and D. Nualart (2010): Parameter estimation for fractional Ornstein-Uhlenbeck processes, Stat. Probab. Lett. 80, no. 11-12, 1030-1038
M. Ledoux: Chaos of a Markov operator and the fourth moment condition, Ann. Probab. 40, no. 6, 2012, 2439-2459
D. Marinucci and I. Wigman: On the Excursion Sets of Spherical Gaussian Eigenfunctions, Journal of Mathematical Physics, 52, 9, 093301, 21 pp. (2011)
D. Marinucci and G. Peccati (2010): Ergodicity and Gaussianity for Spherical Random Fields, J. Math. Phys. 51, 043301
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.
A. Neuenkirch, S. Tindel and J. Unterberger (2010): Discretizing the fractional Levy area, Stoch. Proc. Appl. 120, no. 2, 223-254
I. Nourdin and D. Nualart (2010): Central limit theorems for multiple Skorohod integrals, J. Theoret. Probab. 23, no. 1, 39-64
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 (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 (2010): Universal Gaussian fluctuations of non-Hermitian matrix ensembles: from weak convergence to almost sure CLTs, ALEA 7, 341-375
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, 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, 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.
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 M.S. Taqqu (2010): Wiener Chaos: Moments, Cumulants and Diagrams, Springer Verlag (Bocconi and Springer Series) (see more precisely the chapter 11, entitled ``Limit theorems on the Gaussian Wiener chaos'')
G. Peccati and C. Zheng (2010): Multi-dimensional Gaussian fluctuations on the Poisson space, Elect. J. Probab. 15, no. 48, 1487-1527
A. Réveillac, M. Stauch and C.A. Tudor: Hermite variations of the fractional Brownian sheet. Stoch. Dyn. 12(3), 1150021 (2012), 21 pp
M. Schulte and C. Thaele (2010): Exact and asymptotic results for intrinsic volumes of Poisson k-flat processes

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, M. Podolskij and J. H. C. Woerner (2009): Bipower variation for Gaussian processes with stationary increments, J. Appl. Probab. 46, 132-150
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.
X. Lan and D. Marinucci (2009): On The Dependence Structure of Wavelet Coefficients for Spherical Random Fields, Stoch. Proc. Appl. 119 3749-3766
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 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, G. Peccati and G. Reinert (2009): Second order Poincaré inequalities and CLTs on Wiener space, J. Funct. Anal. 257, 593-609
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 and F.G. Viens (2009): Density formula and concentration inequalities with Malliavin calculus, Electron. J. Probab. 14, 2287-2309
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'')
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.
G. Peccati (2009): Stein's method, Malliavin calculus and infinite-dimensional Gaussian analysis, Progress in Stein’s Method, Singapore
G. Reinert: Gaussian approximation of functionals: Malliavin calculus and Stein’s method, Surveys in stochastic processes, 107–126, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2011
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
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
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.-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
X. Lan and D. Marinucci (2008): The needlets bispectrum, Electron. J. Statist. 2, 332-367
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
I. Nourdin and G. Peccati (2008): Weighted power variations of iterated Brownian motion, Elect. J. Probab. 13, no. 43, 1229-1256
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

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

G. Peccati and C.A. Tudor (2004): Gaussian limits for vector-valued multiple stochastic integrals, Séminaire de Probabilités XXXVIII, 247-262
D. Nualart and G. Peccati (2005): Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. 33, no. 1, 177-193

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