A Berry-Esseen Bound for Nonlinear Statistics with Bounded Differences

  • Nguyen Tien Dung Vietnam National University
  • Hoang Thi Phuong Thao Vietnam National University
  • Pham Trung Hieu Vietnam National University
Keywords: Central limit theorem, Berry-Esseen bound, nonlinear statistic


In this paper, we obtain an explicit Berry-Esseen bound in the central limit theorem for nonlinear statistics with bounded differences. Some examples are provided as well.


V. Bentkus, F. Gotze, A. Tikhomirov, Berry-Esseen bounds for statistics of weakly dependent samples. Bernoulli 3 (1997), no. 3, 329–349.

I. Berkes, W. Philipp, Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7 (1979), no. 1, 29–54.

S. Boucheron, G. Lugosi, P. Massart, Concentration inequalities. A nonasymptotic theory of independence. With a foreword by Michel Ledoux. Oxford University Press, Oxford, 2013.

S. Chatterjee, A new method of normal approximation. Ann. Probab. 36 (2008), no. 4, 1584–1610.

S. Chatterjee, P. Diaconis, A central limit theorem for a new statistic on permutations. Indian J. Pure Appl. Math. 48 (2017), no. 4, 561–573.

L. H. Y. Chen, Q.-M. Shao, Normal approximation for nonlinear statistics using a concentration inequality approach. Bernoulli 13 (2007), no. 2, 581–599.

L. H. Y. Chen, L. Goldstein, Q.-M. Shao, Normal approximation by Stein’s method. Probability and its Applications (New York). Springer, Heidelberg, 2011.

N. T. Dung, Explicit rates of convergence in the multivariate CLT for nonlinear statistics. Acta Math. Hungar. 158 (2019), no. 1, 173–201.

K. O. Friedrich, A Berry-Esseen bound for functions of independent random variables. Ann. Statist. 17 (1989), no. 1, 170–183.

F. Gotze, On the rate of convergence in the multivariate CLT. Ann. Probab. 19 (1991), no. 2, 724–739.

C. Houdre, U. Is¸lak, A Central limit theorem for the length of the longest common subsequences in random words. arXiv:1408.1559v4, 2017.

A. Maurer, M. Pontil, Empirical bounds for functions with weak interactions. Proceedings of Machine Learning Research vol 75:1–24, 2018

R. L-Rey, G. Peccati, New Berry-Esseen bounds for functionals of binomial point processes. Ann. Appl. Probab. 27 (2017), no. 4, 1992–2031.

How to Cite
Tien Dung, N., Thi Phuong Thao, H., & Trung Hieu, P. (2021). A Berry-Esseen Bound for Nonlinear Statistics with Bounded Differences. Statistics, Optimization & Information Computing, 9(4), 984-989. https://doi.org/10.19139/soic-2310-5070-1305
Research Articles