Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Multivariate normal approximation using Stein’s method and Malliavin calculus

Ivan Nourdin, Giovanni Peccati, and Anthony Réveillac

Full-text: Open access

Abstract

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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 46, Number 1 (2010), 45-58.

Dates
First available in Project Euclid: 1 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1267454107

Digital Object Identifier
doi:10.1214/08-AIHP308

Mathematical Reviews number (MathSciNet)
MR2641769

Zentralblatt MATH identifier
1196.60035

Subjects
Primary: 60F05: Central limit and other weak theorems 60G15: Gaussian processes 60H07: Stochastic calculus of variations and the Malliavin calculus

Keywords
Breuer–Major CLT fractional Brownian motion Gaussian processes Malliavin calculus Normal approximation Stein’s method Wasserstein distance

Citation

Nourdin, Ivan; Peccati, Giovanni; Réveillac, Anthony. Multivariate normal approximation using Stein’s method and Malliavin calculus. Ann. Inst. H. Poincaré Probab. Statist. 46 (2010), no. 1, 45--58. doi:10.1214/08-AIHP308. https://projecteuclid.org/euclid.aihp/1267454107


Export citation

References

  • [1] A. D. Barbour. Stein’s method for diffusion approximations. Probab. Theory Related Fields 84 (1990) 297–322.
  • [2] P. Breuer and P. Major. Central limit theorems for nonlinear functionals of Gaussian fields. J. Multivariate Anal. 13 (1983) 425–441.
  • [3] S. Chatterjee. Fluctuation of eigenvalues and second order Poincaré inequalities. Probab. Theory Related Fields 143 (2009) 1–40.
  • [4] S. Chatterjee and E. Meckes. Multivariate normal approximation using exchangeable pairs. ALEA 4 (2008) 257–283.
  • [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.
  • [6] R. M. Dudley. Real Analysis and Probability, 2nd edition. Cambridge Univ. Press, Cambridge, 2003.
  • [7] L. Giraitis and D. Surgailis. CLT and other limit theorems for functionals of Gaussian processes. Z. Wahrsch. Verw. Gebiete 70 (1985) 191–212.
  • [8] F. Götze. On the rate of convergence in the multivariate CLT. Ann. Probab. 19 (1991) 724–739.
  • [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.
  • [10] I. Nourdin and G. Peccati. Non-central convergence of multiple integrals. Ann. Probab. To appear.
  • [11] I. Nourdin and G. Peccati. Stein’s method on Wiener chaos. Probab. Theory Related Fields. To appear.
  • [12] I. Nourdin and G. Peccati. Stein’s method and exact Berry–Esséen asymptotics for functionals of Gaussian fields. Preprint, 2008.
  • [13] D. Nualart. The Malliavin Calculus and Related Topics, 2nd edition. Springer, Berlin, 2006.
  • [14] D. Nualart and S. Ortiz-Latorre. Central limit theorem for multiple stochastic integrals and Malliavin calculus. Stochastic Process. Appl. 118 (2008) 614–628.
  • [15] D. Nualart and G. Peccati. Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33 (2005) 177–193.
  • [16] G. Peccati. Gaussian approximations of multiple integrals. Electron. Comm. Probab. 12 (2007) 350–364 (electronic).
  • [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.
  • [18] G. Reinert and A. Röllin. Multivariate normal approximation with Stein’s method of exchangeable pairs under a general linearity condition. Preprint, 2007.
  • [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.
  • [20] Ch. Stein. Approximate Computation of Expectations. Institute of Mathematical Statistics Lecture Notes, Monograph Series 7. Inst. Math. Statist., Hayward, CA, 1986.