×

zbMATH — the first resource for mathematics

Berry-Esseen bounds and multivariate limit theorems for functionals of Rademacher sequences. (English. French summary) Zbl 1341.60005
Summary: Berry-Esseen bounds for non-linear functionals of infinite Rademacher sequences are derived by means of the Malliavin-Stein method. Moreover, multivariate extensions for vectors of Rademacher functionals are shown. The results establish a connection to small ball probabilities and shed new light onto the relation between central limit theorems on the Rademacher chaos and norms of contraction operators. Applications concern infinite weighted 2-runs, a combinatorial central limit theorem and traces of Bernoulli random matrices.

MSC:
60F05 Central limit and other weak theorems
60F17 Functional limit theorems; invariance principles
60G50 Sums of independent random variables; random walks
60H07 Stochastic calculus of variations and the Malliavin calculus
60B20 Random matrices (probabilistic aspects)
15B52 Random matrices (algebraic aspects)
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] G. W. Anderson and O. Zeitouni. A CLT for a band matrix model. Probab. Theory Related Fields 134 (2) (2006) 283-338. · Zbl 1084.60014
[2] N. Balakrishnan and M. V. Koutras. Runs and Scans with Applications. Wiley Series in Probability and Statistics . Wiley-Interscience [John Wiley & Sons], New York, 2002. · Zbl 0991.62087
[3] R. Blei. Combinatorial dimension and certain norms in harmonic analysis. Amer. J. Math. 106 (4) (1984) 847-887. · Zbl 0579.43010
[4] R. Blei. Analysis in Integer and Fractional Dimensions. Cambridge Studies in Advanced Mathematics 71 . Cambridge Univ. Press, Cambridge, 2001. · Zbl 1006.46001
[5] R. Blei and S. Janson. Rademacher chaos: Tail estimates versus limit theorems. Ark. Mat. 42 (1) (2004) 13-29. · Zbl 1049.60007
[6] S. Chatterjee. A new method of normal approximation. Ann. Probab. 36 (4) (2008) 1584-1610. · Zbl 1159.62009
[7] L. H. Y. Chen, L. Goldstein and Q.-M. Shao. Normal Approximation by Stein’s Method. Probability and Its Applications (New York) . Springer, Heidelberg, 2011.
[8] P. de Jong. A central limit theorem for generalized quadratic forms. Probab. Theory Related Fields 75 (2) (1987) 261-277. · Zbl 0596.60022
[9] V. H. de la Peña and E. Giné. Decoupling: From Dependence to Independence. Probability and Its Applications (New York) . Springer-Verlag, New York, 1999.
[10] P. Eichelsbacher and C. Thäle New Berry-Esseen bounds for non-linear functionals of Poisson random measures. Electron. J. Probab. 19 (102) (2014). · Zbl 1307.60066
[11] A. P. Godbole. The exact and asymptotic distribution of overlapping success runs. Comm. Statist. Theory Methods 21 (4) (1992) 953-967. · Zbl 0800.62068
[12] A. Guionnet. Large Random Matrices: Lectures on Macroscopic Asymptotics. Lecture Notes in Mathematics 1957 . Springer-Verlag, Berlin, 2009.
[13] S. Kwapień and W. A. Woyczyński. Random Series and Stochastic Integrals: Single and Multiple. Probability and Its Applications. Birkhäuser, Boston, MA, 1992.
[14] R. Lachièze-Rey and G. Peccati. Fine Gaussian fluctuations on the Poisson space, I: Contractions, cumulants and geometric random graphs. Electron. J. Probab. 18 (32) (2013) · Zbl 1294.60082
[15] R. Lachièze-Rey and G. Peccati. Fine Gaussian fluctuations on the Poisson space II: Rescaled kernels, marked processes and geometric \(U\)-statistics. Stochastic Process. Appl. 123 (12) (2013) 4186-4218. · Zbl 1294.60082
[16] G. Last, M. D. Penrose, M. Schulte and C. Thäle. Moments and central limit theorems for some multivariate Poisson functionals. Adv. in Appl. Probab. 46 (2) (2014) 348-364. · Zbl 1350.60020
[17] A. E. Litvak, A. Pajor, M. Rudelson and N. Tomczak-Jaegermann. Smallest singular value of random matrices and geometry of random polytopes. Adv. Math. 195 (2) (2005) 491-523. · Zbl 1077.15021
[18] D. Marinucci and G. Peccati. Random Fields on the Sphere. London Mathematical Society Lecture Note Series 389 . Cambridge Univ. Press, Cambridge, 2011. · Zbl 1260.60004
[19] E. Mossel, R. O’Donnell and K. Oleszkiewicz. Noise stability of functions with low influences: Invariance and optimality. Ann. of Math. (2) 171 (1) (2010) 295-341. · Zbl 1201.60031
[20] I. Nourdin and G. Peccati. Stein’s method and exact Berry-Esseen asymptotics for functionals of Gaussian fields. Ann. Probab. 37 (6) (2009) 2231-2261. · Zbl 1196.60034
[21] I. Nourdin and G. Peccati. Stein’s method on Wiener chaos. Probab. Theory Related Fields 145 (1-2) (2009) 75-118. · Zbl 1175.60053
[22] I. Nourdin and G. Peccati. Universal Gaussian fluctuations of non-Hermitian matrix ensembles: From weak convergence to almost sure CLTs. ALEA Lat. Am. J. Probab. Math. Stat. 7 (2010) 341-375. · Zbl 1276.60026
[23] I. Nourdin and G. Peccati. Normal Approximations with Malliavin Calculus: From Stein’s Method to Universality. Cambridge Tracts in Mathematics 192 . Cambridge Univ. Press, Cambridge, MA, 2012. · Zbl 1266.60001
[24] I. Nourdin and G. Peccati. The optimal fourth moment theorem. Proc. Amer. Math. Soc. 143 (7) (2015) 3123-3133. · Zbl 1317.60021
[25] I. Nourdin, G. Peccati and G. Reinert. Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos. Ann. Probab. 38 (5) (2010) 1947-1985. · Zbl 1246.60039
[26] I. Nourdin, G. Peccati and G. Reinert. Stein’s method and stochastic analysis of Rademacher functionals. Electron. J. Probab. 15 (55) (2010) 1703-1742. · Zbl 1225.60046
[27] I. Nourdin, G. Peccati and A. Réveillac. Multivariate normal approximation using Stein’s method and Malliavin calculus. Ann. Inst. Henri Poincaré Probab. Stat. 46 (1) (2010) 45-58. · Zbl 1196.60035
[28] D. Nualart and S. Ortiz-Latorre. Central limit theorems for multiple stochastic integrals and Malliavin calculus. Stochastic Process. Appl. 118 (4) (2008) 614-628. · Zbl 1142.60015
[29] D. Nualart and G. Peccati. Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33 (1) (2005) 177-193. · Zbl 1097.60007
[30] G. Peccati, J. L. Solé and F. Utzet. Stein’s method and normal approximation of Poisson functionals. Ann. Probab. 38 (2) (2010) 443-478. · Zbl 1195.60037
[31] G. Peccati and M. S. Taqqu. Wiener Chaos: Moments, Cumulants and Diagrams. Bocconi & Springer Series 1 . Springer, Bocconi Univ. Press, Milan, 2011. · Zbl 1231.60003
[32] G. Peccati and C. Thäle. Gamma limits and \(U\)-statistics on the Poisson space. ALEA Lat. Am. J. Probab. Math. Stat. 10 (1) (2013) 525-560. · Zbl 1277.60052
[33] 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. · Zbl 1063.60027
[34] G. Peccati and C. Zheng. Multi-dimensional Gaussian fluctuations on the Poisson space. Electron. J. Probab. 15 (48) (2010) 1487-1527. · Zbl 1228.60031
[35] N. Privault. Stochastic analysis of Bernoulli processes. Probab. Surv. 5 (2008) 435-483. · Zbl 1189.60089
[36] G. Reinert and A. Röllin. Multivariate normal approximation with Stein’s method of exchangeable pairs under a general linearity condition. Ann. Probab. 37 (6) (2009) 2150-2173. · Zbl 1200.62010
[37] M. Reitzner and M. Schulte. Central limit theorems for \(U\)-statistics of Poisson point processes. Ann. Probab. 41 (6) (2013) 3879-3909. · Zbl 1293.60061
[38] Y. Rinott and V. Rotar. On coupling constructions and rates in the CLT for dependent summands with applications to the antivoter model and weighted \(U\)-statistics. Ann. Appl. Probab. 7 (4) (1997) 1080-1105. · Zbl 0890.60019
[39] M. Rudelson and R. Vershynin. The Littlewood-Offord problem and invertibility of random matrices. Adv. Math. 218 (2) (2008) 600-633. · Zbl 1139.15015
[40] M. Schulte. A central limit theorem for the Poisson-Voronoi approximation. Adv. in Appl. Math. 49 (3-5) (2012) 285-306. · Zbl 1253.60008
[41] M. Schulte. Normal approximation of Poisson functionals in Kolmogorov distance. J. Theoret. Probab. 29 (1) (2016) 96-117. · Zbl 1335.60027
[42] C. 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) 583-602. Vol. II: Probability Theory . Univ. California Press, Berkeley, CA, 1972.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.