Nourdin, Ivan; Peccati, Giovanni; Réveillac, Anthony
Multivariate normal approximation using Stein's method and Malliavin calculus
Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 1 , p. 45-58
Zbl 1196.60035 | MR 2641769
doi : 10.1214/08-AIHP308
URL stable : http://www.numdam.org/item?id=AIHPB_2010__46_1_45_0

Classification:  60F05,  60G15,  60H07
Nous expliquons comment combiner la méthode de Stein avec les outils du calcul de Malliavin pour majorer, de manière explicite, la distance de Wasserstein entre une fonctionnelle d'un champs gaussien donnée et son approximation normale multidimensionnelle. Entre autres exemples, nous associons des bornes à la version fonctionnelle du théorème de la limite centrale de Breuer-Major, dans le cas du mouvement brownien fractionnaire.
We combine Stein's method with Malliavin calculus in order to obtain explicit bounds in the multidimensional normal approximation (in the Wasserstein distance) of functionals of gaussian fields. Among several examples, we provide an application to a functional version of the Breuer-Major CLT for fields subordinated to a fractional brownian motion.

Bibliographie

[1] A. D. Barbour. Stein's method for diffusion approximations. Probab. Theory Related Fields 84 (1990) 297-322. MR 1035659 | Zbl 0665.60008

[2] P. Breuer and P. Major. Central limit theorems for nonlinear functionals of Gaussian fields. J. Multivariate Anal. 13 (1983) 425-441. MR 716933 | Zbl 0518.60023

[3] S. Chatterjee. Fluctuation of eigenvalues and second order Poincaré inequalities. Probab. Theory Related Fields 143 (2009) 1-40. MR 2449121 | Zbl 1152.60024

[4] S. Chatterjee and E. Meckes. Multivariate normal approximation using exchangeable pairs. ALEA 4 (2008) 257-283. MR 2453473 | Zbl 1162.60310

[5] L. Chen and Q.-M. Shao. Stein's method for normal approximation. In An Introduction to Stein's Method 1-59. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 4. Singapore Univ. Press, Singapore, 2005. MR 2235448

[6] R. M. Dudley. Real Analysis and Probability, 2nd edition. Cambridge Univ. Press, Cambridge, 2003. MR 1932358 | Zbl 1023.60001

[7] L. Giraitis and D. Surgailis. CLT and other limit theorems for functionals of Gaussian processes. Z. Wahrsch. Verw. Gebiete 70 (1985) 191-212. MR 799146 | Zbl 0575.60024

[8] F. Götze. On the rate of convergence in the multivariate CLT. Ann. Probab. 19 (1991) 724-739. MR 1106283 | Zbl 0729.62051

[9] E. P. Hsu. Characterization of Brownian motion on manifolds through integration by parts. In Stein's Method and Applications 195-208. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 5. Singapore Univ. Press, Singapore, 2005. MR 2205337

[10] I. Nourdin and G. Peccati. Non-central convergence of multiple integrals. Ann. Probab. To appear. Zbl 1171.60323

[11] I. Nourdin and G. Peccati. Stein's method on Wiener chaos. Probab. Theory Related Fields. To appear. MR 2520122 | Zbl 1175.60053

[12] I. Nourdin and G. Peccati. Stein's method and exact Berry-Esséen asymptotics for functionals of Gaussian fields. Preprint, 2008. Zbl 1196.60034

[13] D. Nualart. The Malliavin Calculus and Related Topics, 2nd edition. Springer, Berlin, 2006. MR 2200233 | Zbl 0837.60050

[14] D. Nualart and S. Ortiz-Latorre. Central limit theorem for multiple stochastic integrals and Malliavin calculus. Stochastic Process. Appl. 118 (2008) 614-628. MR 2394845 | Zbl 1142.60015

[15] D. Nualart and G. Peccati. Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33 (2005) 177-193. MR 2118863 | Zbl 1097.60007

[16] G. Peccati. Gaussian approximations of multiple integrals. Electron. Comm. Probab. 12 (2007) 350-364 (electronic). MR 2350573 | Zbl 1130.60029

[17] G. Peccati and C. A. Tudor. Gaussian limits for vector-valued multiple stochastic integrals. In Séminaire de Probabilités XXXVIII 247-262. Lecture Notes in Math. 1857. Springer, Berlin, 2005. MR 2126978 | Zbl 1063.60027

[18] G. Reinert and A. Röllin. Multivariate normal approximation with Stein's method of exchangeable pairs under a general linearity condition. Preprint, 2007. Zbl 1200.62010

[19] 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 (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability Theory 583-602. California Univ. Press, Berkeley, CA, 1972. MR 402873 | Zbl 0278.60026

[20] Ch. Stein. Approximate Computation of Expectations. Institute of Mathematical Statistics Lecture Notes, Monograph Series 7. Inst. Math. Statist., Hayward, CA, 1986. MR 882007 | Zbl 0721.60016