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


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

Export citation


  • [1] Arratia, R., Goldstein, L. and Gordon, L. (1990). Poisson approximation and the Chen–Stein method. Statist. Sci. 5 403–424.
  • [2] Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford Studies in Probability 2. Oxford Univ. Press, New York.
  • [3] Cerquetti, A. and Fortini, S. (2006). A Poisson approximation for coloured graphs under exchangeability. Sankhyā 68 183–197.
  • [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.
  • [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.
  • [7] Chatterjee, S. and Dey, P. S. (2010). Applications of Stein’s method for concentration inequalities. Ann. Probab. 38 2443–2485.
  • [8] Chatterjee, S., Diaconis, P. and Meckes, E. (2005). Exchangeable pairs and Poisson approximation. Probab. Surv. 2 64–106.
  • [9] Chatterjee, S. and Meckes, E. (2008). Multivariate normal approximation using exchangeable pairs. ALEA Lat. Am. J. Probab. Math. Stat. 4 257–283.
  • [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.
  • [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.
  • [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.
  • [14] de Jong, P. (1987). A central limit theorem for generalized quadratic forms. Probab. Theory Related Fields 75 261–277.
  • [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.
  • [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.
  • [19] Götze, F. and Tikhomirov, A. (2002). Asymptotic distribution of quadratic forms and applications. J. Theoret. Probab. 15 423–475.
  • [20] Kirkpatrick, K. and Meckes, E. (2013). Asymptotics of the mean-field Heisenberg model. J. Stat. Phys. 152 54–92.
  • [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.
  • [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.
  • [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.
  • [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.
  • [25] Shao, Q. M. and Su, Z. G. (2006). The Berry–Esseen bound for character ratios. Proc. Amer. Math. Soc. 134 2153–2159.
  • [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.
  • [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.
  • [28] Stein, C. (1986). Approximate Computation of Expectations. Institute of Mathematical Statistics Lecture Notes—Monograph Series 7. IMS, Hayward, CA.