## Electronic Journal of Probability

### Quantitative CLTs for symmetric $U$-statistics using contractions

#### Abstract

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 Jammalamadaka and Janson (1986) and Bhattacharaya and Ghosh (1992).

#### Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 5, 43 pp.

Dates
Accepted: 5 January 2019
First available in Project Euclid: 9 February 2019

https://projecteuclid.org/euclid.ejp/1549681361

Digital Object Identifier
doi:10.1214/19-EJP264

Mathematical Reviews number (MathSciNet)
MR3916325

Zentralblatt MATH identifier
07021646

#### Citation

Döbler, Christian; Peccati, Giovanni. Quantitative CLTs for symmetric $U$-statistics using contractions. Electron. J. Probab. 24 (2019), paper no. 5, 43 pp. doi:10.1214/19-EJP264. https://projecteuclid.org/euclid.ejp/1549681361

#### References

• [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.
• [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.
• [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.
• [5] P. de Jong, A central limit theorem for generalized multilinear forms, J. Multivariate Anal. 34 (1990), no. 2, 275–289.
• [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.
• [9] G. G. Gregory, Large sample theory for $U$-statistics and tests of fit, Ann. Statist. 5 (1977), no. 1, 110–123.
• [10] P. Hall, Central limit theorem for integrated square error of multivariate nonparametric density estimators, J. Multivariate Anal. 14 (1984), no. 1, 1–16.
• [11] W. Hoeffding, A class of statistics with asymptotically normal distribution, Ann. Math. Statistics 19 (1948), 293–325.
• [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.
• [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.
• [16] R. Lachièze-Rey and G. Peccati, New Kolmogorov bounds for functionals of binomial point processes, to appear in: Ann. Appl. Probab.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [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.
• [25] G. Peccati and M. Reitzner, Stochastic Analysis for Poisson Point Processes, Mathematics, Statistics, Finance and Economics, Bocconi University Press and Springer, 2016.
• [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.
• [27] G. Peccati and C. Zheng, Multi-dimensional Gaussian fluctuations on the Poisson space, Electron. J. Probab. 15 (2010), no. 48, 1487–1527.
• [28] M. Penrose, Random geometric graphs, Oxford Studies in Probability, vol. 5, Oxford University Press, Oxford, 2003.
• [29] M. Penrose, Geometric random graphs, Oxford, 2004.
• [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.
• [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.
• [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.
• [33] R. J. Serfling, Approximation theorems of mathematical statistics, John Wiley & Sons, Inc., New York, 1980, Wiley Series in Probability and Mathematical Statistics.
• [34] C. Stein, Approximate computation of expectations, Institute of Mathematical Statistics Lecture Notes—Monograph Series, 7, Institute of Mathematical Statistics, Hayward, CA, 1986.
• [35] D. Surgailis, On multiple Poisson stochastic integrals and associated Markov semigroups, Probab. Math. Statist. 3 (1984), no. 2, 217–239.
• [36] R. A. Vitale, Covariances of symmetric statistics, J. Multivariate Anal. 41 (1992), no. 1, 14–26.
• [37] N. C. Weber, Central limit theorems for a class of symmetric statistics, Math. Proc. Cambridge Philos. Soc. 94 (1983), no. 2, 307–313.