Electronic Journal of Probability

Multivariate central limit theorems for Rademacher functionals with applications

Kai Krokowski and Christoph Thäle

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Quantitative multivariate central limit theorems for general functionals of independent, possibly non-symmetric and non-homogeneous infinite Rademacher sequences are proved by combining discrete Malliavin calculus with the smart path method for normal approximation. In particular, a discrete multivariate second-order Poincaré inequality is developed. As a first application, the normal approximation of vectors of subgraph counting statistics in the Erdős-Rényi random graph is considered. In this context, we further specialize to the normal approximation of vectors of vertex degrees. In a second application we prove a quantitative multivariate central limit theorem for vectors of intrinsic volumes induced by random cubical complexes.

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 87, 30 pp.

Received: 25 January 2017
Accepted: 11 September 2017
First available in Project Euclid: 18 October 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 05C80: Random graphs [See also 60B20] 60C05: Combinatorial probability 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60H07: Stochastic calculus of variations and the Malliavin calculus

discrete Malliavin calculus intrinsic volume multivariate central limit theorem smart path method subgraph count random graph random cubical complex vertex degree

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Krokowski, Kai; Thäle, Christoph. Multivariate central limit theorems for Rademacher functionals with applications. Electron. J. Probab. 22 (2017), paper no. 87, 30 pp. doi:10.1214/17-EJP106. https://projecteuclid.org/euclid.ejp/1508292258

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  • [1] Adler, R.J., Bobrowski, O., Borman, M.S., Subag, E. and Weinberger, S.: Persistent homology for random fields and complexes. In Borrowing strength: theory powering applications - a Festschrift for Lawrence D. Brown, IMS Collections 6, 124–143, 2010.
  • [2] Bobrowski, O. and Kahle, M.: Topology of random geometric complexes: a survey. To appear in Topology in Statistical Inference, the Proceedings of Symposia in Applied Mathematics, 2014+.
  • [3] Chatterjee, S.: Fluctuations of eigenvalues and second order Poincaré inequalities. Probab. Theory Relat. Fields 143, (2009), 1–40.
  • [4] Goldstein, L.: A Berry-Esseen bound with applications to vertex degree counts in the Erdős-Rényi random graph. Ann. Appl. Probab. 23, (2013), 617–636.
  • [5] Goldstein, L. and Rinott, Y.: Multivariate normal approximations by Stein’s method and size bias couplings. J. Appl. Probab. 33, (1996), 1–17.
  • [6] Groemer, H.: Eulersche Charakteristik, Projektionen und Quermaßintegrale. Math. Ann. 198, (1972), 23–56.
  • [7] Janson, S. and Nowicki, K.: The asymptotic distributions of generalized U-statistics with applications to random graphs. Probab. Theory Relat. Fields 90, (1991), 341–375.
  • [8] Kahle, M.: Topology of random simplicial complexes: a survey. In Algebraic Topology: Applications and New Directions, Contemporary Mathematics Volume 620, American Mathematical Society, Providence, 201–221, 2014.
  • [9] Krokowski, K.: Poisson approximation of Rademacher functionals by the Chen-Stein method and Malliavin calculus. Commun. Stoch. Anal. 11, (2017), 195–222.
  • [10] Krokowski, K., Reichenbachs, A. and Thäle, C.: Berry-Esseen bounds and multivariate limit theorems for functionals of Rademacher sequences. Ann. Inst. H. Poincaré Probab. Stat. 52, (2016), 763–803.
  • [11] Krokowski, K., Reichenbachs, A. and Thäle, C.: Discrete Malliavin-Stein method: Berry-Esseen bounds for random graphs and percolation. Ann. Probab. 45, (2017), 1071–1109.
  • [12] Linial, N. and Meshulam, R.: Homological connectivity of random 2-complexes. Combinatorica 26, (2006), 475–487.
  • [13] Nourdin, I., Peccati, G. and Reinert, G.: Stein’s method and stochastic analysis of Rademacher functionals. Electron. J. Probab. 15, (2010), 1703–1742.
  • [14] Nourdin, I., Peccati, G. and Réveillac, A.: Multivariate normal approximation using Stein’s method and Malliavin calculus. Ann. Inst. H. Poincaré Probab. Stat. 46, (2010), 45–58.
  • [15] Ohser, J. and Schladitz, K.: 3D Images of Materials Structures: Processing and Analysis. Wiley-VCH Verlag, Weinheim, 2009.
  • [16] Peccati, G. and Zheng, C.: Multi-dimensional Gaussian fluctuations on the Poisson space. Electron. J. Probab. 15, (2010), 1487–1527.
  • [17] Privault, N.: Stochastic Analysis in Discrete and Continuous Settings with Normal Martingales. Lecture Notes in Mathematics 1982, Springer-Verlag, Berlin, 2009.
  • [18] Privault, N. and Torrisi, G.L.: The Stein and Chen-Stein methods for functionals of non-symmetric Bernoulli processes. ALEA Lat. Am. J. Probab. Math. Stat. 12, (2015), 309–356.
  • [19] Raič, M.: CLT-related large deviation bounds based on Stein’s method. Adv. Appl. Probab. 39, (2007), 731–752.
  • [20] Reinert, G. and Röllin, A.: Random subgraph counts and U-statistics: multivariate normal approximation via exchangeable pairs and embedding. J. Appl. Probab. 47, (2010), 378–393.
  • [21] Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory. 2nd Edition, Cambridge University Press, Cambridge, 2013.
  • [22] Svane, A.M.: Estimation of intrinsic volumes from digital grey-scale images. J. Math. Imaging Vision 49, (2016), 352–376.
  • [23] Svane, A.M.: Valuations in image analysis. In Tensor Valuations and their Applications in Stochastic Geometry and Imaging, edited by E.B. Vedel-Jensen and M. Kiderlen, Lecture Notes in Mathematics 2177, Springer-Verlag, Berlin, 435–454, 2017.
  • [24] Talagrand; M.: Spin Glasses: A Challenge for Mathematicians. Springer-Verlag, Berlin, 2003.
  • [25] Werman, M. and Wright, M.L.: Intrinsic volumes of random cubical complexes. Discrete Comput. Geom. 56, (2016), 93–113.
  • [26] Zheng, G.: Normal approximation and almost sure central limit theorem for non-symmetric Rademacher functionals. Stochastic Process. Appl. 127, (2017), 1622–1636.