Stein’s method and exact Berry–Esseen asymptotics for functionals of Gaussian fields

Stein’s method and exact Berry–Esseen asymptotics for functionals of Gaussian fields

Ivan Nourdin and Giovanni Peccati

Source: Ann. Probab. Volume 37, Number 6 (2009), 2231-2261.

Abstract

We show how to detect optimal Berry–Esseen bounds in the normal approximation of functionals of Gaussian fields. Our techniques are based on a combination of Malliavin calculus, Stein’s method and the method of moments and cumulants, and provide de facto local (one-term) Edgeworth expansions. The findings of the present paper represent a further refinement of the main results proven in Nourdin and Peccati [Probab. Theory Related Fields 145 (2009) 75–118]. Among several examples, we discuss three crucial applications: (i) to Toeplitz quadratic functionals of continuous-time stationary processes (extending results by Ginovyan [Probab. Theory Related Fields 100 (1994) 395–406] and Ginovyan and Sahakyan [Probab. Theory Related Fields 138 (2007) 551–579]); (ii) to “exploding” quadratic functionals of a Brownian sheet; and (iii) to a continuous-time version of the Breuer–Major CLT for functionals of a fractional Brownian motion.

First Page: Show

Primary Subjects: 60F05, 60G15, 60H05, 60H07

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

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aop/1258380788
Digital Object Identifier: doi:10.1214/09-AOP461
Mathematical Reviews number (MathSciNet): MR2573557
Zentralblatt MATH identifier: 1196.60034

back to Table of Contents

References

[1] Avram, F. (1988). On bilinear forms in Gaussian random variables and Toeplitz matrices. Probab. Theory Related Fields 79 37–45.

Mathematical Reviews (MathSciNet): MR952991

Zentralblatt MATH: 0648.60043

Digital Object Identifier: doi:10.1007/BF00319101

[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.

Mathematical Reviews (MathSciNet): MR2287104

Zentralblatt MATH: 1107.62038

Digital Object Identifier: doi:10.1016/j.spa.2006.05.015

[3] Breuer, P. and Major, P. (1983). Central limit theorems for nonlinear functionals of Gaussian fields. J. Multivariate Anal. 13 425–441.

Mathematical Reviews (MathSciNet): MR716933

Digital Object Identifier: doi:10.1016/0047-259X(83)90019-2

[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.

Mathematical Reviews (MathSciNet): MR2235448

[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.

Mathematical Reviews (MathSciNet): MR2199561

Zentralblatt MATH: 1090.60020

Digital Object Identifier: doi:10.1016/j.spa.2005.10.004

[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.

Mathematical Reviews (MathSciNet): MR871252

Zentralblatt MATH: 0586.60019

Digital Object Identifier: doi:10.1007/BF00569990

[7] Ginovian, M. S. (1994). On Toeplitz type quadratic functionals of stationary Gaussian processes. Probab. Theory Related Fields 100 395–406.

Mathematical Reviews (MathSciNet): MR1305588

Zentralblatt MATH: 0817.60018

Digital Object Identifier: doi:10.1007/BF01193706

[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.

Mathematical Reviews (MathSciNet): MR2299719

Zentralblatt MATH: 1113.60027

Digital Object Identifier: doi:10.1007/s00440-006-0037-y

[9] Giraitis, L. and Surgailis, D. (1985). CLT and other limit theorems for functionals of Gaussian processes. Z. Wahrsch. Verw. Gebiete 70 191–212.

Mathematical Reviews (MathSciNet): MR799146

[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.

Mathematical Reviews (MathSciNet): MR1061950

Zentralblatt MATH: 0717.62015

Digital Object Identifier: doi:10.1007/BF01207515

[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.

Mathematical Reviews (MathSciNet): MR2363586

[12] Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.

Mathematical Reviews (MathSciNet): MR1145237

[13] Janson, S. (1997). Gaussian Hilbert Spaces. Cambridge Tracts in Mathematics 129. Cambridge Univ. Press, Cambridge.

Mathematical Reviews (MathSciNet): MR1474726

[14] Jeulin, T. (1980). Semi-martingales et Grossissement D’une Filtration. Lecture Notes in Mathematics 833. Springer, Berlin.

Mathematical Reviews (MathSciNet): MR604176

[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.

Mathematical Reviews (MathSciNet): MR544805

[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.

Mathematical Reviews (MathSciNet): MR1232001

[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.

Mathematical Reviews (MathSciNet): MR1863573

Zentralblatt MATH: 01599003

Digital Object Identifier: doi:10.1017/S0266466601171094

JSTOR: links.jstor.org

[18] McCullagh, P. (1987). Tensor Methods in Statistics. Chapman and Hall, London.

Mathematical Reviews (MathSciNet): MR907286

Zentralblatt MATH: 0732.62003

[19] Nourdin, I. and Peccati, G. (2009). Noncentral convergence of multiple integrals. Ann. Probab. 37 1412–1426.

Mathematical Reviews (MathSciNet): MR2546749

Zentralblatt MATH: 1171.60323

Digital Object Identifier: doi:10.1214/08-AOP435

Project Euclid: euclid.aop/1248182142

[20] Nourdin, I. and Peccati, G. (2009). Stein’s method on Wiener chaos. Probab. Theory Related Fields. 145 75–118.

Mathematical Reviews (MathSciNet): MR2520122

Zentralblatt MATH: 05601401

Digital Object Identifier: doi:10.1007/s00440-008-0162-x

[21] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Springer, Berlin.

Mathematical Reviews (MathSciNet): MR2200233

[22] Nualart, D. and Ortiz-Latorre, S. (2008). Central limit theorems for multiple stochastic integrals and Malliavin calculus. Stochastic Process. Appl. 118 614–628.

Mathematical Reviews (MathSciNet): MR2394845

Zentralblatt MATH: 1142.60015

Digital Object Identifier: doi:10.1016/j.spa.2007.05.004

[23] Nualart, D. and Peccati, G. (2005). Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33 177–193.

Mathematical Reviews (MathSciNet): MR2118863

Zentralblatt MATH: 1097.60007

Digital Object Identifier: doi:10.1214/009117904000000621

Project Euclid: euclid.aop/1108141724

[24] Peccati, G. (2001). On the convergence of multiple random integrals. Studia Sci. Math. Hungar. 37 429–470.

Mathematical Reviews (MathSciNet): MR1874695

[25] Peccati, G. (2007). Gaussian approximations of multiple integrals. Electron. Comm. Probab. 12 350–364.

Mathematical Reviews (MathSciNet): MR2350573

Zentralblatt MATH: 1130.60029

[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.

Mathematical Reviews (MathSciNet): MR2126978

Zentralblatt MATH: 1063.60027

[27] Peccati, G. and Yor, M. (2004). Hardy’s inequality in L²([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.

Mathematical Reviews (MathSciNet): MR2106849

[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.

Mathematical Reviews (MathSciNet): MR2235451

[30] Rota, G.-C. and Wallstrom, T. C. (1997). Stochastic integrals: A combinatorial approach. Ann. Probab. 25 1257–1283.

Mathematical Reviews (MathSciNet): MR1457619

Zentralblatt MATH: 0886.60046

Digital Object Identifier: doi:10.1214/aop/1024404513

Project Euclid: euclid.aop/1024404513

[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.

Mathematical Reviews (MathSciNet): MR2201886

[32] Shigekawa, I. (1978). Absolute continuity of probability laws of Wiener functionals. Proc. Japan Acad. Ser. A Math. Sci. 54 230–233.

Mathematical Reviews (MathSciNet): MR517327

Digital Object Identifier: doi:10.3792/pjaa.54.230

Project Euclid: euclid.pja/1195517581

[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.

Mathematical Reviews (MathSciNet): MR402873

Zentralblatt MATH: 0278.60026

[34] Stein, C. (1986). Approximate Computation of Expectations. IMS, Hayward, CA.

Mathematical Reviews (MathSciNet): MR882007

Zentralblatt MATH: 0721.60016

[35] Taniguchi, M. (1986). Berry–Esseen theorems for quadratic forms of Gaussian stationary processes. Probab. Theory Related Fields 72 185–194.

Mathematical Reviews (MathSciNet): MR836274

Zentralblatt MATH: 0572.60030

Digital Object Identifier: doi:10.1007/BF00699102

previous :: next