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Quantitative CLTs for symmetric \(U\)-statistics using contractions. (English) Zbl 1442.60034
Summary: We consider sequences of symmetric \(U\)-statistics, not necessarily Hoeffding-degenerate, both in a one- and multi-dimensional setting, and prove quantitative central limit theorems (CLTs) based on the use of contraction operators. Our results represent an explicit counterpart to analogous criteria that are available for sequences of random variables living on the Gaussian, Poisson or Rademacher chaoses, and are perfectly tailored for geometric applications. As a demonstration of this fact, we develop explicit bounds for subgraph counting in generalised random graphs on Euclidean spaces; special attention is devoted to the so-called ’dense parameter regime’ for uniformly distributed points, for which we deduce CLTs that are new even in their qualitative statement, and that substantially extend classical findings by S. R. Jammalamadaka and S. Janson [Ann. Probab. 14, 1347–1358 (1986; Zbl 0604.60023)] and R. N. Bhattacharya and J. K. Ghosh [J. Multivariate Anal. 43, No. 2, 300–330 (1992; Zbl 0764.60025)].

60F05 Central limit and other weak theorems
60D05 Geometric probability and stochastic geometry
62G99 Nonparametric inference
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[1] R. N. Bhattacharya and J. K. Ghosh, A class of \(U\)-statistics and asymptotic normality of the number of \(k\)-clusters, J. Multivariate Anal. 43 (1992), no. 2, 300–330. · Zbl 0764.60025
[2] S. Bourguin and G. Peccati, The Malliavin-Stein method on the Poisson space, Stochastic analysis for Poisson point processes (G. Peccati and M. Reitzner, eds.), Mathematics, Statistics, Finance and Economics, Bocconi University Press and Springer, 2016, pp. 185–228.
[3] S. Chatterjee and E. Meckes, Multivariate normal approximation using exchangeable pairs, ALEA Lat. Am. J. Probab. Math. Stat. 4 (2008), 257–283. · Zbl 1162.60310
[4] P. de Jong, Central limit theorems for generalized multilinear forms, CWI Tract, vol. 61, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1989. · Zbl 0677.60029
[5] P. de Jong, A central limit theorem for generalized multilinear forms, J. Multivariate Anal. 34 (1990), no. 2, 275–289. · Zbl 0709.60019
[6] C. Döbler, New developments in Stein’s method with applications, (2012), (Ph.D.)-Thesis Ruhr-Universität Bochum.
[7] C. Döbler and G. Peccati, Quantiative de Jong theorems in any dimension, Electron. J. Probab. 22 (2017), no. 2, 1–35.
[8] E. B. Dynkin and A. Mandelbaum, Symmetric statistics, Poisson point processes, and multiple Wiener integrals, Ann. Statist. 11 (1983), no. 3, 739–745. · Zbl 0518.60050
[9] G. G. Gregory, Large sample theory for \(U\)-statistics and tests of fit, Ann. Statist. 5 (1977), no. 1, 110–123. · Zbl 0371.62033
[10] P. Hall, Central limit theorem for integrated square error of multivariate nonparametric density estimators, J. Multivariate Anal. 14 (1984), no. 1, 1–16. · Zbl 0528.62028
[11] W. Hoeffding, A class of statistics with asymptotically normal distribution, Ann. Math. Statistics 19 (1948), 293–325. · Zbl 0032.04101
[12] S. R. Jammalamadaka and S. Janson, Limit theorems for a triangular scheme of \(U\)-statistics with applications to inter-point distances, Ann. Probab. 14 (1986), no. 4, 1347–1358. · Zbl 0604.60023
[13] V. S. Koroljuk and Yu. V. Borovskich, Theory of \(U\)-statistics, Mathematics and its Applications, vol. 273, Kluwer Academic Publishers Group, Dordrecht, 1994, Translated from the 1989 Russian original by P. V. Malyshev and D. V. Malyshev and revised by the authors.
[14] K. Krokowski, Poisson approximation of Rademacher functionals by the Chen-Stein method and Malliavin calculus, Commun. Stoch. Anal. 11 (2017), no. 2, 195–222.
[15] K. Krokowski, A. Reichenbachs, and C. Thäle, Berry-Esseen bounds and multivariate limit theorems for functionals of Rademacher sequences, Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016), no. 2, 763–803. · Zbl 1341.60005
[16] R. Lachièze-Rey and G. Peccati, New Kolmogorov bounds for functionals of binomial point processes, to appear in: Ann. Appl. Probab. · Zbl 1374.60023
[17] R. Lachièze-Rey and G. Peccati, Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs, Electron. J. Probab. 18 (2013), no. 32, 32. · Zbl 1295.60015
[18] 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 (2013), no. 12, 4186–4218. · Zbl 1294.60082
[19] G. Last, Stochastic analysis for Poisson processes, Stochastic analysis for Poisson point processes (G. Peccati and M. Reitzner, eds.), Mathematics, Statistics, Finance and Economics, Bocconi University Press and Springer, 2016, pp. 1–36.
[20] P. Major, On the estimation of multiple random integrals and \(U\)-statistics, Lecture Notes in Mathematics, vol. 2079, Springer, Heidelberg, 2013. · Zbl 1280.60002
[21] E. Meckes, On Stein’s method for multivariate normal approximation, High dimensional probability V: the Luminy volume, Inst. Math. Stat. Collect., vol. 5, Inst. Math. Statist., Beachwood, OH, 2009, pp. 153–178. · Zbl 1243.60025
[22] I. Nourdin and G. Peccati, Normal approximations with Malliavin calculus, Cambridge Tracts in Mathematics, vol. 192, Cambridge University Press, Cambridge, 2012, From Stein’s method to universality. · Zbl 1266.60001
[23] I. Nourdin, G. Peccati, and G. Reinert, Invariance principles for homogeneous sums: universality of Gaussian Wiener chaos, Ann. Probab. 38 (2010), no. 5, 1947–1985. · Zbl 1246.60039
[24] I. Nourdin, G. Peccati, and G. Reinert, Stein’s method and stochastic analysis of Rademacher functionals, Electron. J. Probab. 15 (2010), no. 55, 1703–1742. · Zbl 1225.60046
[25] G. Peccati and M. Reitzner, Stochastic Analysis for Poisson Point Processes, Mathematics, Statistics, Finance and Economics, Bocconi University Press and Springer, 2016. · Zbl 1350.60005
[26] G. Peccati, J. L. Solé, M. S. Taqqu, and F. Utzet, Stein’s method and normal approximation of Poisson functionals, Ann. Probab. 38 (2010), no. 2, 443–478. · Zbl 1195.60037
[27] G. Peccati and C. Zheng, Multi-dimensional Gaussian fluctuations on the Poisson space, Electron. J. Probab. 15 (2010), no. 48, 1487–1527. · Zbl 1228.60031
[28] M. Penrose, Random geometric graphs, Oxford Studies in Probability, vol. 5, Oxford University Press, Oxford, 2003. · Zbl 1029.60007
[29] M. Penrose, Geometric random graphs, Oxford, 2004. · Zbl 1029.60007
[30] N. Privault and G. L. Torrisi, The Stein and Chen-Stein methods for functionals of non-symmetric Bernoulli processes, ALEA Lat. Am. J. Probab. Math. Stat. 12 (2015), no. 1, 309–356. · Zbl 1329.60079
[31] G. Reinert and A. Röllin, Random subgraph counts and \(U\)-statistics: multivariate normal approximation via exchangeable pairs and embedding, J. Appl. Probab. 47 (2010), no. 2, 378–393. · Zbl 1210.62009
[32] 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 (1997), no. 4, 1080–1105. · Zbl 0890.60019
[33] R. J. Serfling, Approximation theorems of mathematical statistics, John Wiley & Sons, Inc., New York, 1980, Wiley Series in Probability and Mathematical Statistics. · Zbl 0538.62002
[34] C. Stein, Approximate computation of expectations, Institute of Mathematical Statistics Lecture Notes—Monograph Series, 7, Institute of Mathematical Statistics, Hayward, CA, 1986. · Zbl 0721.60016
[35] D. Surgailis, On multiple Poisson stochastic integrals and associated Markov semigroups, Probab. Math. Statist. 3 (1984), no. 2, 217–239. · Zbl 0548.60058
[36] R. A. Vitale, Covariances of symmetric statistics, J. Multivariate Anal. 41 (1992), no. 1, 14–26. · Zbl 0759.62021
[37] N. C. Weber, Central limit theorems for a class of symmetric statistics, Math. Proc. Cambridge Philos. Soc. 94 (1983), no. 2, 307–313. · Zbl 0563.60025
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