Stein's method and exact Berry-Esseen asymptotics for functionals of Gaussian fields

Annals of Probability

Volumn 37, Issue 6, 2009, Pages 2231-2261

  Nourdin, Iva ^a   Peccati, Giovanni ^b  

^a University of Paris VII LPSM partially supported by ERC Starting Grant 680275 MALIG   (France)

^b UNIVERSITÉ PARIS OUEST NANTERRE LA DÉFENSE   (France)

Author keywords

Berry Esseen bounds; Breuer Major CLT; Brownian sheet; Fractional Brownian motion; Local Edgeworth expansions; Malliavin calculus; Multiple stochastic integrals; Normal approximation; Optimal rates; Quadratic functionals; Stein's method; Toeplitz quadratic forms

Indexed keywords

EID: 77049091106     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/09-AOP461     Document Type: Article

References (35)

1. Avram, F. (1988). On bilinear forms in Gaussian random variables and Toeplitz matrices. Probab. Theory Related Fields 79 37-45.
2. Bhansali, R.J., Giraitis, L.and Kokoszka, P. S. (2007). Approximations and limit theory for quadratic forms of linear processes. Stochastic Process. Appl. 117 71-95.
3. Breuer, P. and Major, P. (1983). Central limit theorems for nonlinear functionals of Gaussian fields. J. Multivariate Anal. 13 425-441.
4. 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.
5. Deheuvels, P., Peccati, G. and Yor, M. (2006). On quadratic functionals of the Brownian sheet and related processes. Stochastic Process. Appl. 116 493-538.
6. Fox, R. and Taqqu, M. S. (1987). Central limit theorems for quadratic forms in random variables having long-range dependence. Probab. Theory Related Fields 74 213-240.
7. Ginovian, M. S. (1994). On Toeplitz type quadratic functionals of stationary Gaussian processes. Probab. Theory Related Fields 100 395-406.
8. Ginovyan, M.S. and Sahakyan, A. A. (2007). Limit theorems for Toeplitz quadratic functionals of continuous-time stationary processes. Probab. Theory Related Fields 138 551-579.
9. Giraitis, L. and Surgailis, D. (1985). CLT and other limit theorems for functionals of Gaussian processes. Z. Wahrsch. Verw. Gebiete 70 191-212.
10. Giraitis, L. and Surgailis, D. (1990). A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotical normality of Whittle's estimate. Probab. Theory Related Fields 86 87-104.
11. Götze, F., Tikhomirov, A. and Yurchenko, V. (2007). Asymptotic expansion in the central limit theorem for quadratic forms. J. Math. Sci. 147 6891-6911.
12. Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New york.
13. Janson, S. (1997). Gaussian Hilbert Spaces. Cambridge Tracts in Mathematics 129.Cambridge Univ. Press, Cambridge.
14. Jeulin, T. (1980). Semi-martingales et Grossissement D'une Filtration. Lecture Notes in Mathematics 833. Springer, Berlin.
15. Jeulin, T. and Yor, M. (1979). Inégalité de Hardy, semimartingales, et faux-amis. In Séminaire de Probabilités XIII. Lecture Notes in Math. 721 332-359. Springer, Berlin.
16. Jeulin, T. and Yor, M. (1992). Une décomposition non-canonique du drap brownien. In Séminaire de Probabilités, XXVI. Lecture Notes in Math. 1526 322-347. Springer, Berlin.
17. Lieberman, O., Rousseau, J.and Zucker, D. M. (2001). Valid Edgeworth expansion for the sample autocorrelation function under long range dependence. Econometric Theory 17 257-275.
18. Mccullagh, P. (1987). Tensor Methods in Statistics. Chapman and Hall, London.
19. Nourdin, I. and Peccati, G. (2009). Noncentral convergence of multiple integrals. Ann. Probab. 37 1412-1426.
20. Nourdin, I. and Peccati, G. (2009). Stein's method on Wiener chaos. Probab. Theory Related Fields. 145 75-118.
21. Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Springer, Berlin.
22. Nualart, D.and Ortiz-Latorre, S. (2008). Central limit theorems for multiple stochastic integrals and Malliavin calculus. Stochastic Process. Appl. 118 614-628.
23. Nualart, D.and Peccati, G. (2005). Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33 177-193.
24. Peccati, G. (2001). On the convergence of multiple random integrals. Studia Sci. Math. Hungar. 37 429-470.
25. Peccati, G. (2007). Gaussian approximations of multiple integrals. Electron. Comm. Probab. 12 350-364.
26. 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.
27. 2([0, 1]) and principal values of Brownian local times. In Asymptotic Methods in Stochastics. Fields Inst. Commun. 44 49-74. Amer. Math. Soc., Providence, RI.
28. Peccati, G.and Yor, M. (2004). Four limit theorems for quadratic functionals of Brownian motion and Brownian bridge. In Asymptotic Methods in Stochastics. Fields Inst. Commun. 44 75-87. Amer. Math. Soc., Providence, RI.
29. 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.
30. Rota, G.-C.and Wallstrom, T. C. (1997). Stochastic integrals: A combinatorial approach. Ann. Probab. 25 1257-1283.
31. Rotar, V. (2005). Stein's method, Edgeworth's expansions and a formula of Barbour. In Stein's Method and Applications. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 5 59-84. Singapore Univ. Press, Singapore.
32. Shigekawa, I. (1978). Absolute continuity of probability laws of Wiener functionals. Proc. Japan Acad. Ser. AMath. Sci. 54 230-233.
33. 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 Symp. Math. Statist. Probab. Vol. II: Probability Theory 583 602. Univ. California Press, Berkeley. MR0402873
34. STEIN, C. (1986). Approximate Computation of Expectations. IMS, Hayward, CA. MR882007
35. TANIGUCHI, M. (1986). Berry Esseen theorems for quadratic forms of Gaussian stationary processes. Probab. Theory Related Fields 72 185 194. MR836274