Invariance principles for homogeneous sums: Universality of gaussian wiener chaos

Annals of Probability

Volumn 38, Issue 5, 2010, Pages 1947-1985

  Nourdin, Ivan ^a   Peccati, Giovanni ^b,c   Reinert, Gesine ^d  

^a SORBONNE UNIVERSITÉ   (France)

^b UNIVERSITY OF LUXEMBOURG   (Luxembourg)

^c UNIVERSITÉ PARIS OUEST NANTERRE LA DÉFENSE   (France)

^d UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Central limit theorems; Chaos; Chi square limit theorems; Homogeneous sums; Lindeberg principle; Malliavin calculus; Stein's method; Universality; Wiener chaos

Indexed keywords

EID: 77957150726     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/10-AOP531     Document Type: Article

References (31)

1. CHEN, L. H. Y. and SHAO, Q.-M. (2005). Stein's method for normal approximation. In An Introduction to Stein's Method. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 4 1-59. Singapore Univ. Press, Singapore.
2. DAVIDOV, Y. and ROTAR', V. (2009). On asymptotic proximity of distributions. J. Theoret. Probab. 22 82-98.
3. DE JONG, P. (1989). Central Limit Theorems for Generalized Multilinear Forms. CWI Tract 61. Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam.
4. DE JONG, P. (1990). A central limit theorem for generalized multilinear forms. J. Multivariate Anal. 34 275-289.
5. GöTZE, F. (1991). On the rate of convergence in the multivariate CLT. Ann. Probab. 19 724-739.
6. JANSON, S. (1997). Gaussian Hilbert Spaces. Cambridge Tracts in Mathematics 129. Cambridge Univ. Press, Cambridge.
7. LOH, W.-L. (2008). A multivariate central limit theorem for randomized orthogonal array sampling designs in computer experiments. Ann. Statist. 36 1983-2023.
8. MALLIAVIN, P. (1997). Stochastic Analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 313. Springer, Berlin.
9. MOSSEL, E. (2010). Gaussian bounds for noise correlation of functions. GAFA 19 1713-1756.
10. MOSSEL, E., O'DONNELL, R. and OLESZKIEWICZ, K. (2010). Noise stability of functions with low influences: Variance and optimality. Ann. Math. 171 295-341.
11. NOURDIN, I. and PECCATI, G. (2009). Non-central convergence of multiple integrals. Ann. Probab. 37 14121426.
12. NOURDIN, I. and PECCATI, G. (2009). Stein's method on Wiener chaos. Probab. Theory Related Fields 145 75-118.
13. NOURDIN, I. and PECCATI, G. (2009). Stein's method and exact Berry-Esseen asymptotics for functionals of Gaussian fields. Ann. Probab. 37 2231-2261.
14. NOURDIN, I. and PECCATI, G. (2009). Universal Gaussian fluctuations of non-Hermitian matrix ensembles. Preprint.
15. NOURDIN, I., PECCATI, G. and REINERT, G. (2008). Stein's method and stochastic analysis of Rademacher functionals. Preprint.
16. NOURDIN, I., PECCATI, G. and REINERT, G. (2009). Second order Poincaré inequalities and CLTs on Wiener space. J. Funct. Anal. 257 593-609.
17. NOURDIN, I., PECCATI, G. and RÉVEILLAC, A. (2010). Multivariate normal approximation using Stein's method and Malliavin calculus. Ann. Inst. H. Poincaré Probab. Statist. 46 45-58.
18. NUALART, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Springer, Berlin.
19. NUALART, D. and ORTIZ-LATORRE, S. (2008). Central limit theorems for multiple stochastic integrals and Malliavin calculus. Stochastic Process. Appl. 118 614-628.
20. NUALART, D. and PECCATI, G. (2005). Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33 177-193.
21. PECCATI, G., SOLÉ, J. L., TAQQU, M. S. and UTZET, F. (2010). Stein's method and normal approximation of Poisson functionals. Ann. Probab. 38 443-478.
22. PECCATI, G. and TAQQU, M. S. (2008). Moments, cumulants and diagram formulae for nonlinear functionals of random measure (Survey). Preprint.
23. PECCATI, G. and TUDOR, C. A. (2005). Gaussian limits for vector-valued multiple stochastic integrals. In Séminaire de Probabilités XXXVIII. Lecture Notes in Math. 1857 247-262. Springer, Berlin.
24. PRIVAULT, N. (2009). Stochastic Analysis in Discrete and Continuous Settings with Normal Martingales. Lecture Notes in Math. 1982. Springer, Berlin.
25. REINERT, G. (2005). Three general approaches to Stein's method. In An Introduction to Stein's Method. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 4 183-221. Singapore Univ. Press, Singapore.
26. -1/2 log n rate and applications to multivariate graph related statistics. J. Multivariate Anal. 56 333-350.
27. ROTAR', V. I. (1975). Limit theorems for multilinear forms and quasipolynomial functions. Teor. Verojatnost. i Primenen. 20 527-546.
28. ROTAR', V. I. (1979). Limit theorems for polylinear forms. J. Multivariate Anal. 9 511-530.
29. STEIN, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proc. Sixth Berkeley Sympos. Math. Statist. Probab., Vol. II: Probability Theory 583-602. Univ. California Press, Berkeley.
30. STEIN, C. (1986). Approximate Computation of Expectations. Institute of Mathematical Statistics Lecture Notes-Monograph Series 7. IMS, Hayward, CA.
31. TAO, T. and VU, V. (2008). Random matrices: The circular law. Commun. Contemp. Math. 10 261-307.