## Electronic Journal of Probability

### Quantitative de Jong theorems in any dimension

#### Abstract

We develop a new quantitative approach to a multidimensional version of the well-known de Jong’s central limit theorem under optimal conditions, stating that a sequence of Hoeffding degenerate $U$-statistics whose fourth cumulants converge to zero satisfies a CLT, as soon as a Lindeberg-Feller type condition is verified. Our approach allows one to deduce explicit (and presumably optimal) Wasserstein bounds in the case of general $U$-statistics of arbitrary order $d\geq 1$. One of our main findings is that, for vectors of $U$-statistics satisfying de Jong’ s conditions and whose covariances admit a limit, componentwise convergence systematically implies joint convergence to Gaussian: this is the first instance in which such a phenomenon is described outside the frameworks of homogeneous chaoses and of diffusive Markov semigroups.

#### Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 2, 35 pp.

Dates
Accepted: 15 December 2016
First available in Project Euclid: 5 January 2017

https://projecteuclid.org/euclid.ejp/1483585524

Digital Object Identifier
doi:10.1214/16-EJP19

Mathematical Reviews number (MathSciNet)
MR3613695

Zentralblatt MATH identifier
1357.60023

#### Citation

Döbler, Christian; Peccati, Giovanni. Quantitative de Jong theorems in any dimension. Electron. J. Probab. 22 (2017), paper no. 2, 35 pp. doi:10.1214/16-EJP19. https://projecteuclid.org/euclid.ejp/1483585524

#### References

• [1] O. Arizmendi, Convergence of the fourth moment and infinite divisibility, Probab. Math. Statist. 33 (2013), no. 2, 201–212.
• [2] E. Azmoodeh, S. Campese, and G. Poly, Fourth Moment Theorems for Markov diffusion generators, J. Funct. Anal. 266 (2014), no. 4, 2341–2359.
• [3] E. Bolthausen, Exact convergence rates in some martingale central limit theorems, Ann. Probab. 10 (1982), no. 3, 672–688.
• [4] S. Bourguin and G. Peccati, Portmanteau inequalities on the Poisson space: mixed regimes and multidimensional clustering, Electron. J. Probab. 19 (2014), no. 66, 42.
• [5] S. Bourguin and G. Peccati, Semicircular limits on the free Poisson chaos: counterexamples to a transfer principle, J. Funct. Anal. 267 (2014), no. 4, 963–997.
• [6] S. Campese, I. Nourdin, G. Peccati, and G. Poly, Multivariate Gaussian approximations on Markov chaoses, Electron. Commun. Probab. 21 (2016), Paper No. 48, 9.
• [7] S. Chatterjee, J. Fulman, and A. Röllin, Exponential approximation by Stein’s method and spectral graph theory, ALEA Lat. Am. J. Probab. Math. Stat. 8 (2011), 197–223.
• [8] S. Chatterjee and E. Meckes, Multivariate normal approximation using exchangeable pairs, ALEA Lat. Am. J. Probab. Math. Stat. 4 (2008), 257–283.
• [9] S. Chatterjee and Q.-M. Shao, Nonnormal approximation by Stein’s method of exchangeable pairs with application to the Curie-Weiss model, Ann. Appl. Probab. 21 (2011), no. 2, 464–483.
• [10] 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.
• [11] P. de Jong, Central limit theorems for generalized multilinear forms, CWI Tract, vol. 61, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1989.
• [12] P. de Jong, A central limit theorem for generalized multilinear forms, J. Multivariate Anal. 34 (1990), no. 2, 275–289.
• [13] C. Döbler, New developments in Stein’s method with applications, (2012), (Ph.D.)-Thesis Ruhr-Universität Bochum.
• [14] C. Döbler, Stein’s method of exchangeable pairs for the Beta distribution and generalizations, Electron. J. Probab. 20 (2015), no. 109, 1–34.
• [15] E. B. Dynkin and A. Mandelbaum, Symmetric statistics, Poisson point processes, and multiple Wiener integrals, Ann. Statist. 11 (1983), no. 3, 739–745.
• [16] P. Eichelsbacher and M. Löwe, Stein’s method for dependent random variables occurring in statistical mechanics, Electron. J. Probab. 15 (2010), no. 30, 962–988.
• [17] P. Eichelsbacher and C. Thäle, New Berry-Esseen bounds for non-linear functionals of Poisson random measures, Electron. J. Probab. 19 (2014), no. 102, 25.
• [18] O. El-Dakkak and G. Peccati, Hoeffding decompositions and urn sequences, Ann. Probab. 36 (2008), no. 6, 2280–2310.
• [19] O. El-Dakkak, G. Peccati, and I. Prünster, Exchangeable Hoeffding decompositions over finite sets: a combinatorial characterization and counterexamples, J. Multivariate Anal. 131 (2014), 51–64.
• [20] T. Fissler and C. Thäle, A four moments theorem for gamma limits on a Poisson chaos, ALEA Lat. Am. J. Probab. Math. Stat. 13 (2016), no. 1, 163–192.
• [21] J. Fulman and N. Ross, Exponential approximation and Stein’s method of exchangeable pairs, ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013), no. 1, 1–13.
• [22] G. G. Gregory, Large sample theory for $U$-statistics and tests of fit, Ann. Statist. 5 (1977), no. 1, 110–123.
• [23] E. Haeusler, On the rate of convergence in the central limit theorem for martingales with discrete and continuous time, Ann. Probab. 16 (1988), no. 1, 275–299.
• [24] C. C. Heyde and B. M. Brown, On the departure from normality of a certain class of martingales, Ann. Math. Statist. 41 (1970), 2161–2165.
• [25] W. Hoeffding, A class of statistics with asymptotically normal distribution, Ann. Math. Statistics 19 (1948), 293–325.
• [26] 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.
• [27] S. Janson, Gaussian Hilbert spaces, Cambridge Tracts in Mathematics, vol. 129, Cambridge University Press, Cambridge, 1997.
• [28] O. Kallenberg, Foundations of modern probability, second ed., Probability and its Applications (New York), Springer-Verlag, New York, 2002.
• [29] S. Karlin and Y. Rinott, Applications of ANOVA type decompositions for comparisons of conditional variance statistics including jackknife estimates, Ann. Statist. 10 (1982), no. 2, 485–501.
• [30] T. Kemp, G. Nourdin, I.and Peccati, and R. Speicher, Wigner chaos and the fourth moment, Ann. Probab. 40 (2012), no. 4, 1577–1635.
• [31] 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.
• [32] K. Krokowski, A. Reichenbachs, and C. Thäle, Discrete Malliavin-Stein method: Berry-Esseen bounds for random graphs and percolation, to appear in: Ann. Probab.
• [33] R. Lachièze-Rey and G. Peccati, New Kolmogorov bounds for functionals of binomial point processes, to appear in: Ann. Appl. Probab.
• [34] 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.
• [35] 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.
• [36] M. Ledoux, Chaos of a Markov operator and the fourth moment condition, Ann. Probab. 40 (2012), no. 6, 2439–2459.
• [37] W. G. McGinley and R. Sibson, Dissociated random variables, Math. Proc. Cambridge Philos. Soc. 77 (1975), 185–188.
• [38] 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.
• [39] E. Mossel, R. O’Donnell, and K. Oleszkiewicz, Noise stability of functions with low influences: invariance and optimality, Ann. of Math. (2) 171 (2010), no. 1, 295–341.
• [40] I. Nourdin and G. Peccati, Stein’s method on Wiener chaos, Probab. Theory Related Fields 145 (2009), no. 1–2, 75–118.
• [41] 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.
• [42] I. Nourdin, G. Peccati, G. Poly, and R. Simone, Multidimensional limit theorems for homogeneous sums: a survey and a general transfer principle, ESAIM Probab. Stat. 20 (2016), 293–308.
• [43] 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.
• [44] 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.
• [45] I. Nourdin, G. Peccati, and R. Speicher, Multi-dimensional semicircular limits on the free Wigner chaos, Seminar on Stochastic Analysis, Random Fields and Applications VII, Progr. Probab., vol. 67, Birkhäuser/Springer, Basel, 2013, pp. 211–221.
• [46] D. Nualart and G. Peccati, Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. 33 (2005), no. 1, 177–193.
• [47] G. Peccati, Hoeffding-ANOVA decompositions for symmetric statistics of exchangeable observations, Ann. Probab. 32 (2004), no. 3A, 1796–1829.
• [48] G. Peccati and M. Reitzner (eds.), Stochastic analysis for poisson point processes, Springer-Verlag, 2016.
• [49] 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.
• [50] G. Peccati and C. Thäle, Gamma limits and $U$-statistics on the Poisson space, ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013), no. 1, 525–560.
• [51] G. Peccati and C. A. Tudor, Gaussian limits for vector-valued multiple stochastic integrals, Séminaire de Probabilités XXXVIII, Lecture Notes in Math., vol. 1857, Springer, Berlin, 2005, pp. 247–262.
• [52] G. Peccati and C. Zheng, Multi-dimensional Gaussian fluctuations on the Poisson space, Electron. J. Probab. 15 (2010), no. 48, 1487–1527.
• [53] G. Peccati and C. Zheng, Universal Gaussian fluctuations on the discrete Poisson chaos, Bernoulli 20 (2014), no. 2, 697–715.
• [54] M. Penrose, Random geometric graphs, Oxford Studies in Probability, vol. 5, Oxford University Press, Oxford, 2003.
• [55] 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.
• [56] G. Reinert and A. Röllin, Multivariate normal approximation with Stein’s method of exchangeable pairs under a general linearity condition, Ann. Probab. 37 (2009), no. 6, 2150–2173.
• [57] M. Reitzner and M. Schulte, Central limit theorems for $U$-statistics of Poisson point processes, Ann. Probab. 41 (2013), no. 6, 3879–3909.
• [58] 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.
• [59] A. Röllin, A note on the exchangeability condition in Stein’s method, Statist. Probab. Lett. 78 (2008), no. 13, 1800–1806.
• [60] H. Rubin and R. A. Vitale, Asymptotic distribution of symmetric statistics, Ann. Statist. 8 (1980), no. 1, 165–170.
• [61] M. Schulte, Normal Approximation of Poisson Functionals in Kolmogorov Distance, J. Theoret. Probab. 29 (2016), no. 1, 96–117.
• [62] R. J. Serfling, Approximation theorems of mathematical statistics, John Wiley & Sons, Inc., New York, 1980, Wiley Series in Probability and Mathematical Statistics.
• [63] C. Stein, Approximate computation of expectations, Institute of Mathematical Statistics Lecture Notes—Monograph Series, 7, Institute of Mathematical Statistics, Hayward, CA, 1986.
• [64] R. A. Vitale, Covariances of symmetric statistics, J. Multivariate Anal. 41 (1992), no. 1, 14–26.