Bernoulli

  • Bernoulli
  • Volume 21, Number 1 (2015), 335-359.

On the error bound in a combinatorial central limit theorem

Louis H.Y. Chen and Xiao Fang

Full-text: Open access

Abstract

Let $\mathbb{X}=\{X_{ij}\colon\ 1\le i,j\le n\}$ be an $n\times n$ array of independent random variables where $n\ge2$. Let $\pi$ be a uniform random permutation of $\{1,2,\dots,n\}$, independent of $\mathbb{X}$, and let $W=\sum_{i=1}^{n}X_{i\pi(i)}$. Suppose $\mathbb{X}$ is standardized so that $\mathbb{E}W=0$, $\operatorname{Var}(W)=1$. We prove that the Kolmogorov distance between the distribution of $W$ and the standard normal distribution is bounded by $451\sum_{i,j=1}^{n}\mathbb{E}|X_{ij}|^{3}/n$. Our approach is by Stein’s method of exchangeable pairs and the use of a concentration inequality.

Article information

Source
Bernoulli, Volume 21, Number 1 (2015), 335-359.

Dates
First available in Project Euclid: 17 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.bj/1426597072

Digital Object Identifier
doi:10.3150/13-BEJ569

Mathematical Reviews number (MathSciNet)
MR3322321

Zentralblatt MATH identifier
1354.60011

Keywords
combinatorial central limit theorem concentration inequality exchangeable pairs Stein’s method

Citation

Chen, Louis H.Y.; Fang, Xiao. On the error bound in a combinatorial central limit theorem. Bernoulli 21 (2015), no. 1, 335--359. doi:10.3150/13-BEJ569. https://projecteuclid.org/euclid.bj/1426597072


Export citation

References

  • [1] Barbour, A.D. and Chen, L.H.Y. (2005). An Introduction to Stein’s Method. Lecture Notes Series 4. Singapore: Institute for Mathematical Sciences, National Univ. Singapore, Singapore Univ. Press and World Scientific.
  • [2] Bolthausen, E. (1984). An estimate of the remainder in a combinatorial central limit theorem. Z. Wahrsch. Verw. Gebiete 66 379–386.
  • [3] Chen, L.H.Y. (1975/76). An approximation theorem for sums of certain randomly selected indicators. Z. Wahrsch. Verw. Gebiete 33 69–74.
  • [4] Chen, L.H.Y. (1986). The Rate of Convergence in a Central Limit Theorem for Dependent Random Variables with Arbitrary Index Set. IMA Preprint Series 243. Minneapolis: Univ. Minnesota.
  • [5] Chen, L.H.Y. (1998). Stein’s method: Some perspectives with applications. In Probability Towards 2000 (L. Accardi and C.C. Heyde, eds.). Lecture Notes in Statistics 128. New York: Springer. Papers from the symposium held at Columbia Univ., New York, October 2–6, 1995.
  • [6] Chen, L.H.Y., Goldstein, L. and Shao, Q.-M. (2011). Normal Approximation by Stein’s Method. Probability and Its Applications (New York). Heidelberg: Springer.
  • [7] Chen, L.H.Y. and Shao, Q.-M. (2001). A non-uniform Berry–Esseen bound via Stein’s method. Probab. Theory Related Fields 120 236–254.
  • [8] Chen, L.H.Y. and Shao, Q.-M. (2004). Normal approximation under local dependence. Ann. Probab. 32 1985–2028.
  • [9] Chen, L.H.Y. and Shao, Q.-M. (2005). Stein’s method for normal approximation. In An Introduction to Stein’s Method (A.D. Barbour and L.H.Y. Chen, eds.). Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 4 1–59. Singapore: Singapore Univ. Press.
  • [10] Diaconis, P. (1977). The distribution of leading digits and uniform distribution $\mathrm{mod}$ $1$. Ann. Probab. 5 72–81.
  • [11] Goldstein, L. (2005). Berry–Esseen bounds for combinatorial central limit theorems and pattern occurrences, using zero and size biasing. J. Appl. Probab. 42 661–683.
  • [12] Goldstein, L. (2007). $L^{1}$ bounds in normal approximation. Ann. Probab. 35 1888–1930.
  • [13] Ho, S.T. and Chen, L.H.Y. (1978). An $L_{p}$ bound for the remainder in a combinatorial central limit theorem. Ann. Probab. 6 231–249.
  • [14] Hoeffding, W. (1951). A combinatorial central limit theorem. Ann. Math. Statistics 22 558–566.
  • [15] Johnson, W.B. and Lindenstrauss, J. (1984). Extensions of Lipschitz mappings into a Hilbert space. In Conference in Modern Analysis and Probability (New Haven, Conn., 1982). Contemp. Math. 26 189–206. Providence, RI: Amer. Math. Soc.
  • [16] Neammanee, K. and Rattanawong, P. (2008). A uniform bound on the generalization of a combinatorial central limit theorem. Int. Math. Forum 3 11–27.
  • [17] Neammanee, K. and Rerkruthairat, N. (2012). An improvement of a uniform bound on a combinatorial central limit theorem. Comm. Statist. Theory Methods 41 1590–1602.
  • [18] Neammanee, K. and Suntornchost, J. (2005). A uniform bound on a combinatorial central limit theorem. Stoch. Anal. Appl. 23 559–578.
  • [19] Rinott, Y. and Rotar, V. (1997). 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 1080–1105.
  • [20] Shao, Q.-M. and Su, Z.-G. (2006). The Berry–Esseen bound for character ratios. Proc. Amer. Math. Soc. 134 2153–2159.
  • [21] Stein, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability Theory 583–602. Berkeley, CA: Univ. California Press.
  • [22] Stein, C. (1986). Approximate Computation of Expectations. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 7. Hayward, CA: IMS.
  • [23] Wald, A. and Wolfowitz, J. (1944). Statistical tests based on permutations of the observations. Ann. Math. Statistics 15 358–372.
  • [24] Wolff, P. (2012). On randomness reduction in the Johnson–Lindenstrauss lemma. Preprint, available at http://arxiv.org/abs/1202.5500.