The Annals of Probability

From Stein identities to moderate deviations

Louis H. Y. Chen, Xiao Fang, and Qi-Man Shao

Full-text: Open access

Abstract

Stein’s method is applied to obtain a general Cramér-type moderate deviation result for dependent random variables whose dependence is defined in terms of a Stein identity. A corollary for zero-bias coupling is deduced. The result is also applied to a combinatorial central limit theorem, a general system of binary codes, the anti-voter model on a complete graph, and the Curie–Weiss model. A general moderate deviation result for independent random variables is also proved.

Article information

Source
Ann. Probab., Volume 41, Number 1 (2013), 262-293.

Dates
First available in Project Euclid: 23 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1358951987

Digital Object Identifier
doi:10.1214/12-AOP746

Mathematical Reviews number (MathSciNet)
MR3059199

Zentralblatt MATH identifier
1275.60029

Subjects
Primary: 60F10: Large deviations
Secondary: 60F05: Central limit and other weak theorems

Keywords
Stein’s method Stein identity moderate deviations Berry–Esseen bounds zero-bias coupling exchangeable pairs dependent random variables combinatorial central limit theorem general system of binary codes anti-voter model Curie–Weiss model

Citation

Chen, Louis H. Y.; Fang, Xiao; Shao, Qi-Man. From Stein identities to moderate deviations. Ann. Probab. 41 (2013), no. 1, 262--293. doi:10.1214/12-AOP746. https://projecteuclid.org/euclid.aop/1358951987


Export citation

References

  • Baldi, P., Rinott, Y. and Stein, C. (1989). A normal approximation for the number of local maxima of a random function on a graph. In Probability, Statistics, and Mathematics 59–81. Academic Press, Boston, MA.
  • Barbour, A. D. (1988). Stein’s method and Poisson process convergence. J. Appl. Probab. 25A 175–184.
  • Barbour, A. D. (1990). Stein’s method for diffusion approximations. Probab. Theory Related Fields 84 297–322.
  • Barbour, A. D. and Chen, L. H. Y. (2005). An Introduction to Stein Method. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 4. Singapore Univ. Press and World Scientific, Singapore.
  • Chatterjee, S. (2007). Stein’s method for concentration inequalities. Probab. Theory Related Fields 138 305–321.
  • Chatterjee, S. (2008). A new method of normal approximation. Ann. Probab. 36 1584–1610.
  • Chatterjee, S. and Dey, P. S. (2010). Applications of Stein’s method for concentration inequalities. Ann. Probab. 38 2443–2485.
  • 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.
  • Chen, L. H. Y. (1975). Poisson approximation for dependent trials. Ann. Probab. 3 534–545.
  • Chen, L. H. Y., Hwang, H. K. and Zacharovas, V. (2011). Distribution of the sum-of-digits function of random integers: A survey. Unpublished manuscript.
  • Chen, L. H. Y. and Röllin, A. (2010). Stein couplings for normal approximation. Preprint. Available at http://arxiv.org/abs/1003.6039.
  • 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.
  • Chen, L. H. Y. and Shao, Q.-M. (2004). Normal approximation under local dependence. Ann. Probab. 32 1985–2028.
  • 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 Univ. Press and World Scientific, Singapore.
  • Cramér, H. (1938). Sur un nouveau théorème-limite de la théorie des probabilités. Actualités Scientifiques et Industrielles 736 5–23.
  • Dembo, A. and Rinott, Y. (1996). Some examples of normal approximations by Stein’s method. In Random Discrete Structures (Minneapolis, MN, 1993). The IMA Volumes in Mathematics and its Applications 76 25–44. Springer, New York.
  • Diaconis, P. (1977). The distribution of leading digits and uniform distribution mod 1. Ann. Probab. 5 72–81.
  • Diaconis, P. and Holmes, S. (2004). Stein’s Method: Expository Lectures and Applications. Institute of Mathematical Statistics Lecture Notes 46. IMS, Hayward, CA.
  • Ellis, R. S. (1985). Entropy, Large Deviations, and Statistical Mechanics. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 271. Springer, New York.
  • Ellis, R. S. and Newman, C. M. (1978a). The statistics of Curie–Weiss models. J. Stat. Phys. 19 149–161.
  • Ellis, R. S. and Newman, C. M. (1978b). Limit theorems for sums of dependent random variables occurring in statistical mechanics. Z. Wahrsch. Verw. Gebiete 44 117–139.
  • 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.
  • Goldstein, L. and Reinert, G. (1997). Stein’s method and the zero bias transformation with application to simple random sampling. Ann. Appl. Probab. 7 935–952.
  • Linnik, J. V. (1961). On the probability of large deviations for the sums of independent variables. In Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II 289–306. Univ. California Press, Berkeley, CA.
  • Nourdin, I. and Peccati, G. (2009). Stein’s method on Wiener chaos. Probab. Theory Related Fields 145 75–118.
  • Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, New York.
  • Raič, M. (2007). CLT-related large deviation bounds based on Stein’s method. Adv. in Appl. Probab. 39 731–752.
  • 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.
  • Shao, Q.-M. (2010). Stein’s method, self-normalized limit theory and applications. In Proceedings of the International Congress of Mathematicians. Volume IV 2325–2350. Hindustan Book Agency, New Delhi.
  • 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. Univ. California Press, Berkeley, CA.
  • Stein, C. (1986). Approximate Computation of Expectations. Institute of Mathematical Statistics Lecture Notes—Monograph Series 7. IMS, Hayward, CA.