The Annals of Applied Probability

New Berry–Esseen bounds for functionals of binomial point processes

Raphaël Lachièze-Rey and Giovanni Peccati

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We obtain explicit Berry–Esseen bounds in the Kolmogorov distance for the normal approximation of nonlinear functionals of vectors of independent random variables. Our results are based on the use of Stein’s method and of random difference operators, and generalise the bounds obtained by Chatterjee (2008), concerning normal approximations in the Wasserstein distance. In order to obtain lower bounds for variances, we also revisit the classical Hoeffding decompositions, for which we provide a new proof and a new representation. Several applications are discussed in detail: in particular, new Berry–Esseen bounds are obtained for set approximations with random tessellations, as well as for functionals of coverage processes.

Article information

Ann. Appl. Probab., Volume 27, Number 4 (2017), 1992-2031.

Received: May 2015
Revised: January 2016
First available in Project Euclid: 30 August 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Berry–Esseen bounds binomial processes covering processes random tessellations stochastic geometry Stein’s method


Lachièze-Rey, Raphaël; Peccati, Giovanni. New Berry–Esseen bounds for functionals of binomial point processes. Ann. Appl. Probab. 27 (2017), no. 4, 1992--2031. doi:10.1214/16-AAP1218.

Export citation


  • [1] Boucheron, S., Lugosi, G. and Massart, P. (2013). Concentration Inequalities: A Nonasymptotic Theory of Independence. Oxford Univ. Press, Oxford.
  • [2] Bourguin, S. and Peccati, G. (2016). Stein and Chen–Stein methods and Malliavin calculus on the Poisson space. In Stochastic Analysis for Poisson Point Processes: Malliavin Calculus, Wiener–Itô Chaos Expansions and Stochastic Geometry. Springer, Berlin.
  • [3] Calka, P. and Chenavier, N. (2014). Extreme values for characteristic radii of a Poisson–Voronoi tessellation. Extremes 17 359–385.
  • [4] Chatterjee, S. (2008). A new method of normal approximation. Ann. Probab. 36 1584–1610.
  • [5] Chatterjee, S. (2014). Superconcentration and Related Topics. Springer, Cham.
  • [6] Chen, L. H. Y., Goldstein, L. and Shao, Q. M. (2011). Normal Approximation by Stein’s Method. Springer, Heidelberg.
  • [7] Cuevas, A., Fraiman, R. and Rodríguez-Casal, A. (2007). A nonparametric approach to the estimation of lengths and surface areas. Ann. Statist. 35 1031–1051.
  • [8] Eichelsbacher, P. and Thäle, C. (2014). New Berry–Esseen bounds for non-linear functionals of Poisson random measures. Electron. J. Probab. 19 no. 102, 25.
  • [9] Einmahl, J. H. J. and Khmaladze, E. V. (2001). The two-sample problem in $\mathbb{R}^{m}$ and measure-valued martingales. In State of the Art in Probability and Statistics (Leiden, 1999). Institute of Mathematical Statistics Lecture Notes—Monograph Series 36 434–463. IMS, Beachwood, OH.
  • [10] Federer, H. (1959). Curvature measures. Trans. Amer. Math. Soc. 93 418–491.
  • [11] Gloria, A. and Nolen, J. (2015). A quantitative central limit theorem for the effective conductance on the discrete torus. Comm. Pure Appl. Math. DOI:10.1002/cpa.21614.
  • [12] Goldstein, L. and Penrose, M. D. (2010). Normal approximation for coverage models over binomial point processes. Ann. Appl. Probab. 20 696–721.
  • [13] Gong, R., Houdré, C. and Islak, Ü. (2015). A central limit theorem for the optimal alignments score in multiple random words. Preprint.
  • [14] Heveling, M. and Reitzner, M. (2009). Poisson–Voronoi approximation. Ann. Appl. Probab. 19 719–736.
  • [15] Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution. Ann. Math. Stat. 19 293–325.
  • [16] Houdré, C. and Islak, Ü. (2014). A central limit theorem for the length of the longest common subsequence in random words. Preprint. Available at arXiv:1408.1559.
  • [17] Karlin, S. and Rinott, Y. (1982). Applications of ANOVA type decompositions for comparisons of conditional variance statistics including jackknife estimates. Ann. Statist. 10 485–501.
  • [18] Kendall, W. S. and Molchanov, I. (2010). New Perspectives in Stochastic Geometry. Oxford Univ. Press, Oxford. Edited by Wilfrid S. Kendall and Ilya Molchanov.
  • [19] Lachièze-Rey, R. and Vega, S. (2015). Boundary density and Voronoi approximation of irregular sets. Trans. Amer. Math. Soc. 369 4953–4976.
  • [20] Last, G., Peccati, G. and Schulte, M. (2015). Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization. Probab. Theory Related Fields 165 667–723.
  • [21] Molchanov, I. (1997). Statistics of the Boolean Model for Practitioners and Mathematicians. Wiley, New York.
  • [22] Molchanov, I. (2005). Theory of Random Sets. Springer, London.
  • [23] Nolen, J. (2015). Normal approximation for the net flux through a random conductor. J. Stoch. PDE Anal. Comp. 4 439–476.
  • [24] Peccati, G. (2004). Hoeffding-ANOVA decompositions for symmetric statistics of exchangeable observations. Ann. Probab. 32 1796–1829.
  • [25] Reitzner, M., Spodarev, E. and Zaporozhets, D. (2012). Set reconstruction by Voronoi cells. Adv. in Appl. Probab. 44 938–953.
  • [26] Rhee, W. T. and Talagrand, M. (1986). Martingale inequalities and the jackknife estimate of variance. Statist. Probab. Lett. 4 5–6.
  • [27] Rodríguez Casal, A. (2007). Set estimation under convexity type assumptions. Ann. Inst. Henri Poincaré Probab. Stat. 43 763–774.
  • [28] Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.
  • [29] Schulte, M. (2016). Normal approximation of Poisson functionals in Kolmogorov distance. J. Theoret. Probab. 29 96–117.
  • [30] Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. Wiley, New York.
  • [31] Thäle, C. and Yukich, J. E. (2016). Asymptotic theory for statistics of the Poisson–Voronoi approximation. Bernoulli 22 2372–2400.
  • [32] Vitale, R. A. (1992). Covariances of symmetric statistics. J. Multivariate Anal. 41 14–26.
  • [33] Walther, G. (1999). On a generalization of Blaschke’s rolling theorem and the smoothing of surfaces. Math. Methods Appl. Sci. 22 301–316.
  • [34] Yukich, J. E. (2015). Surface order scaling in stochastic geometry. Ann. Appl. Probab. 25 177–210.