Electronic Journal of Probability

Wasserstein-2 bounds in normal approximation under local dependence

Xiao Fang

Full-text: Open access

Abstract

We obtain a general bound for the Wasserstein-2 distance in normal approximation for sums of locally dependent random variables. The proof is based on an asymptotic expansion for expectations of second-order differentiable functions of the sum. We apply the main result to obtain Wasserstein-2 bounds in normal approximation for sums of $m$-dependent random variables, U-statistics and subgraph counts in the Erdős-Rényi random graph. We state a conjecture on Wasserstein-$p$ bounds for any positive integer $p$ and provide supporting arguments for the conjecture.

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 35, 14 pp.

Dates
Received: 1 February 2019
Accepted: 24 March 2019
First available in Project Euclid: 9 April 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1554775414

Digital Object Identifier
doi:10.1214/19-EJP301

Mathematical Reviews number (MathSciNet)
MR3940765

Zentralblatt MATH identifier
1412.60043

Subjects
Primary: 60F05: Central limit and other weak theorems

Keywords
central limit theorem local dependence Erdős-Rényi random graph Stein’s method U-statistics Wasserstein-2 distance

Rights
Creative Commons Attribution 4.0 International License.

Citation

Fang, Xiao. Wasserstein-2 bounds in normal approximation under local dependence. Electron. J. Probab. 24 (2019), paper no. 35, 14 pp. doi:10.1214/19-EJP301. https://projecteuclid.org/euclid.ejp/1554775414


Export citation

References

  • [1] Barbour, A. D. (1986). Asymptotic expansions based on smooth functions in the central limit theorem. Probab. Theory Relat. Fields 72, no. 2, 289–303.
  • [2] Barbour, A. D., Karoński, M. and Ruciński, A. (1989). A central limit theorem for decomposable random variables with applications to random graphs. J. Combin. Theory Ser. B 47, no. 2, 125–145.
  • [3] Bobkov, S. G. (2018). Berry-Esseen bounds and Edgeworth expansions in the central limit theorem for transport distances. Probab. Theory Related Fields 170, no. 1-2, 229–262.
  • [4] Bonis, T. (2018). Rate in the central limit theorem and diffusion approximation via Stein’s method. Preprint. Available at https://arxiv.org/abs/1506.06966
  • [5] Chen, L.H.Y., Goldstein, L. and Shao, Q.M. (2011). Normal approximation by Stein’s method. Probability and its Applications (New York). Springer, Heidelberg, 2011. xii+405 pp.
  • [6] Chen, L.H.Y. and Shao, Q.M. (2004). Normal approximation under local dependence. Ann. Probab. 32, no. 3A, 1985–2028.
  • [7] Chen, L.H.Y. and Shao, Q.M. (2007). Normal approximation for nonlinear statistics using a concentration inequality approach. Bernoulli 13, 581–599.
  • [8] Courtade, T.A., Fathi, M. and Pananjady, A. (2018). Existence of Stein kernels under a spectral gap, and discrepancy bound. Preprint. Available at https://arxiv.org/abs/1703.07707
  • [9] Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution. Ann. Math. Statistics 19, 293–325.
  • [10] Ledoux, M., Nourdin, I. and Peccati, G. (2015). Stein’s method, logarithmic Sobolev and transport inequalities. Geom. Funct. Anal. 25, no. 1, 256–306.
  • [11] Rinott, Y. and Rotar, V. (2003). On Edgeworth expansions for dependency-neighborhoods chain structures and Stein’s method. Probab. Theory Related Fields 126, no. 4, 528–570.
  • [12] Rio, E. (2009). Upper bounds for minimal distances in the central limit theorem. Ann. Inst. Henri Poincaré Probab. Stat. 45, no. 3, 802–817.
  • [13] Röllin, A. (2017). Kolmogorov bounds for the normal approximation of the number of triangles in the Erdös-Rényi random graph. Preprint. Available at https://arxiv.org/abs/1704.00410
  • [14] Stein, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Proc. Sixth Berkeley Symp. Math. Stat. Prob. 2, Univ. California Press. Berkeley, Calif., 583–602.
  • [15] Zhai, A. (2018). A high-dimensional CLT in $\mathcal{W} _2$ distance with near optimal convergence rate. Probab. Theory Related Fields 170, no. 3-4, 821–845.