The Annals of Probability

Berry–Esseen bounds of normal and nonnormal approximation for unbounded exchangeable pairs

Qi-Man Shao and Zhuo-Song Zhang

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

Abstract

An exchangeable pair approach is commonly taken in the normal and nonnormal approximation using Stein’s method. It has been successfully used to identify the limiting distribution and provide an error of approximation. However, when the difference of the exchangeable pair is not bounded by a small deterministic constant, the error bound is often not optimal. In this paper, using the exchangeable pair approach of Stein’s method, a new Berry–Esseen bound for an arbitrary random variable is established without a bound on the difference of the exchangeable pair. An optimal convergence rate for normal and nonnormal approximation is achieved when the result is applied to various examples including the quadratic forms, general Curie–Weiss model, mean field Heisenberg model and colored graph model.

Article information

Source
Ann. Probab., Volume 47, Number 1 (2019), 61-108.

Dates
Received: November 2016
Revised: December 2017
First available in Project Euclid: 13 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.aop/1544691618

Digital Object Identifier
doi:10.1214/18-AOP1255

Mathematical Reviews number (MathSciNet)
MR3909966

Zentralblatt MATH identifier
07036334

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Stein’s method exchangeable pairs Berry–Esseen bound quadratic forms simple random sampling general Curie–Weiss model mean field Heisenberg model monochromatic edges

Citation

Shao, Qi-Man; Zhang, Zhuo-Song. Berry–Esseen bounds of normal and nonnormal approximation for unbounded exchangeable pairs. Ann. Probab. 47 (2019), no. 1, 61--108. doi:10.1214/18-AOP1255. https://projecteuclid.org/euclid.aop/1544691618


Export citation

References

  • [1] Arratia, R., Goldstein, L. and Gordon, L. (1990). Poisson approximation and the Chen–Stein method. Statist. Sci. 5 403–424.
    Zentralblatt MATH: 0955.62542
    Digital Object Identifier: doi:10.1214/ss/1177012015
    Project Euclid: euclid.ss/1177012015
  • [2] Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford Studies in Probability 2. Oxford Univ. Press, New York.
    Zentralblatt MATH: 0746.60002
  • [3] Cerquetti, A. and Fortini, S. (2006). A Poisson approximation for coloured graphs under exchangeability. Sankhyā 68 183–197.
    Zentralblatt MATH: 1192.60053
  • [4] Chatterjee, S. (2005). Concentration inequalities with exchangeable pairs. Ph.D. thesis, Stanford Univ., Stanford, CA.
  • [5] Chatterjee, S. (2008). A new method of normal approximation. Ann. Probab. 36 1584–1610.
    Zentralblatt MATH: 1159.62009
    Digital Object Identifier: doi:10.1214/07-AOP370
    Project Euclid: euclid.aop/1217360979
  • [6] Chatterjee, S. (2014). A short survey of Stein’s method. In Proceedings of the International Congress of Mathematicians—Seoul 2014 (S. Y. Jang, Y. R. Kim, D.-W. Lee and I. Yie, eds.) IV 1–24.
    Zentralblatt MATH: 1373.60052
  • [7] Chatterjee, S. and Dey, P. S. (2010). Applications of Stein’s method for concentration inequalities. Ann. Probab. 38 2443–2485.
    Zentralblatt MATH: 1203.60023
    Digital Object Identifier: doi:10.1214/10-AOP542
    Project Euclid: euclid.aop/1285334211
  • [8] Chatterjee, S., Diaconis, P. and Meckes, E. (2005). Exchangeable pairs and Poisson approximation. Probab. Surv. 2 64–106.
    Zentralblatt MATH: 1189.60072
    Digital Object Identifier: doi:10.1214/154957805100000096
  • [9] Chatterjee, S. and Meckes, E. (2008). Multivariate normal approximation using exchangeable pairs. ALEA Lat. Am. J. Probab. Math. Stat. 4 257–283.
    Zentralblatt MATH: 1162.60310
  • [10] Chatterjee, S. and Shao, Q. M. (2011). Nonnormal approximation by Stein’s method of exchangeable pairs with application to the Curie–Weiss model. Ann. Appl. Probab. 21 464–483.
    Zentralblatt MATH: 1216.60018
    Digital Object Identifier: doi:10.1214/10-AAP712
    Project Euclid: euclid.aoap/1300800979
  • [11] Chen, L. H. Y., Fang, X. and Shao, Q. M. (2013). Moderate deviations in Poisson approximation: A first attempt. Statist. Sinica 23 1523–1540.
    Zentralblatt MATH: 06232419
  • [12] Chen, L. H. Y., Goldstein, L. and Shao, Q. M. (2011). Normal Approximation by Stein’s Method. Springer, Heidelberg.
  • [13] Chen, Y. and Shao, Q. M. (2012). Berry–Esséen inequality for unbounded exchangeable pairs. In Probability Approximations and Beyond (A. Barbour, H. P. Chan and D. Siegmund, eds.). Lecture Notes in Statistics 205 13–30. Springer, New York.
    Mathematical Reviews (MathSciNet): MR3289373
    Digital Object Identifier: doi:10.1007/978-1-4614-1966-2
  • [14] de Jong, P. (1987). A central limit theorem for generalized quadratic forms. Probab. Theory Related Fields 75 261–277.
    Zentralblatt MATH: 0596.60022
    Digital Object Identifier: doi:10.1007/BF00354037
  • [15] Ellis, R. and Newman, C. (1978). Limit theorems for sums of dependent random variables occurring in statistical mechanics. Z. Wahrsch. Verw. Gebiete 44 117–139.
    Zentralblatt MATH: 0364.60120
    Digital Object Identifier: doi:10.1007/BF00533049
  • [16] Ellis, R. S. and Newman, C. M. (1978). The statistics of Curie–Weiss models. J. Stat. Phys. 19 149–161.
  • [17] Ellis, R. S. and Newman, C. M. (1978). Fluctuationes in Curie–Weiss exemplis. In Mathematical Problems in Theoretical Physics (Proc. Internat. Conf., Univ. Rome, Rome, 1977). Lecture Notes in Physics 80 313–324. Springer, Berlin–New York.
  • [18] Fang, X. (2015). A universal error bound in the CLT for counting monochromatic edges in uniformly colored graphs. Electron. Commun. Probab. 20 1–6.
    Zentralblatt MATH: 1320.05039
    Digital Object Identifier: doi:10.1214/ECP.v20-3707
  • [19] Götze, F. and Tikhomirov, A. (2002). Asymptotic distribution of quadratic forms and applications. J. Theoret. Probab. 15 423–475.
    Zentralblatt MATH: 1002.60018
    Digital Object Identifier: doi:10.1023/A:1014867011101
  • [20] Kirkpatrick, K. and Meckes, E. (2013). Asymptotics of the mean-field Heisenberg model. J. Stat. Phys. 152 54–92.
    Zentralblatt MATH: 1281.82007
    Digital Object Identifier: doi:10.1007/s10955-013-0753-5
  • [21] Meckes, E. (2009). On Stein’s method for multivariate normal approximation. In High Dimensional Probability V: The Luminy Volume 158–178. IMS, Beachwood, OH.
    Zentralblatt MATH: 1243.60025
  • [22] Reinert, G. and Röllin, A. (2009). Multivariate normal approximation with Stein’s method of exchangeable pairs under a general linearity condition. Ann. Probab. 37 2150–2173.
    Mathematical Reviews (MathSciNet): MR2573554
    Zentralblatt MATH: 1200.62010
    Digital Object Identifier: doi:10.1214/09-AOP467
    Project Euclid: euclid.aop/1258380785
  • [23] Rinott, Y. and Rotar, V. (1996). A multivariate CLT for local dependence with $n^{-1/2}\log n$ rate and applications to multivariate graph related statistics. J. Multivariate Anal. 56 333–350.
    Zentralblatt MATH: 0859.60019
    Digital Object Identifier: doi:10.1006/jmva.1996.0017
  • [24] 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.
    Zentralblatt MATH: 0890.60019
    Digital Object Identifier: doi:10.1214/aoap/1043862425
    Project Euclid: euclid.aoap/1043862425
  • [25] Shao, Q. M. and Su, Z. G. (2006). The Berry–Esseen bound for character ratios. Proc. Amer. Math. Soc. 134 2153–2159.
    Zentralblatt MATH: 1093.60014
    Digital Object Identifier: doi:10.1090/S0002-9939-05-08177-3
  • [26] Shao, Q. M., Zhang, M. C. and Zhang, Z. S. (2017). Cramér type moderate deviations for non-normal approximation. Available at arXiv:1809.07966.
    arXiv: 1809.07966
  • [27] Shao, Q. M. and Zhang, Z. S. (2016). Identifying the limiting distribution by a general non-normal approximation of Stein’s method. Sci. China Math. 59 2379–2392.
    Zentralblatt MATH: 1362.60020
    Digital Object Identifier: doi:10.1007/s11425-016-0322-3
  • [28] Stein, C. (1986). Approximate Computation of Expectations. Institute of Mathematical Statistics Lecture Notes—Monograph Series 7. IMS, Hayward, CA.
    Zentralblatt MATH: 0721.60016

The Institute of Mathematical Statistics

The Annals of Probability

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more