previous :: next
Applications of Stein’s method for concentration inequalities
Sourav Chatterjee and Partha S. Dey
Source: Ann. Probab. Volume 38, Number 6
(2010), 2443-2485.
Abstract
Stein’s method for concentration inequalities was introduced to prove concentration of measure in problems involving complex dependencies such as random permutations and Gibbs measures. In this paper, we provide some extensions of the theory and three applications: (1) We obtain a concentration inequality for the magnetization in the Curie–Weiss model at critical temperature (where it obeys a nonstandard normalization and super-Gaussian concentration). (2) We derive exact large deviation asymptotics for the number of triangles in the Erdős–Rényi random graph G(n, p) when p ≥ 0.31. Similar results are derived also for general subgraph counts. (3) We obtain some interesting concentration inequalities for the Ising model on lattices that hold at all temperatures.
First Page:
Show
Hide
Keywords: Stein’s method; Gibbs measures; concentration inequality; Ising model; Curie–Weiss model; large deviation; Erdős–Rényi random graph; exponential random graph
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
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1285334211
Digital Object Identifier: doi:10.1214/10-AOP542
Mathematical Reviews number (MathSciNet): MR2683635
Zentralblatt MATH identifier: 1203.60023
References
[1] Barthe, F., Cattiaux, P. and Roberto, C. (2005). Concentration for independent random variables with heavy tails. AMRX Appl. Math. Res. Express 2 39–60.
Mathematical Reviews (MathSciNet): MR2173316
Zentralblatt MATH: 1094.60010
Digital Object Identifier: doi:10.1155/AMRX.2005.39
[2] Barthe, F., Cattiaux, P. and Roberto, C. (2006). Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry. Rev. Mat. Iberoamericana 22 993–1067.
Mathematical Reviews (MathSciNet): MR2320410
Project Euclid: euclid.rmi/1169480039
[3] Bhamidi, S., Bresler, G. and Sly, A. (2008). Mixing time of exponential random graphs. In Proc. of the 49th Annual IEEE Symp. on FOCS 803–812. IEEE Computer Society, Washington, DC.
[4] Bobkov, S. G. (2007). Large deviations and isoperimetry over convex probability measures with heavy tails. Electron. J. Probab. 12 1072–1100 (electronic).
Mathematical Reviews (MathSciNet): MR2336600
Zentralblatt MATH: 1148.60011
[5] Bobkov, S. G. and Ledoux, M. (2000). From Brunn–Minkowski to Brascamp–Lieb and to logarithmic Sobolev inequalities. Geom. Funct. Anal. 10 1028–1052.
Mathematical Reviews (MathSciNet): MR1800062
Zentralblatt MATH: 0969.26019
Digital Object Identifier: doi:10.1007/PL00001645
[6] Bollobás, B. (2001). Random Graphs, 2nd ed. Cambridge Studies in Advanced Mathematics 73. Cambridge Univ. Press, Cambridge.
[7] Bolthausen, E. (1987). Laplace approximations for sums of independent random vectors. II. Degenerate maxima and manifolds of maxima. Probab. Theory Related Fields 76 167–206.
Mathematical Reviews (MathSciNet): MR906774
Zentralblatt MATH: 0608.60018
Digital Object Identifier: doi:10.1007/BF00319983
[8] Bolthausen, E., Comets, F. and Dembo, A. (2009). Large deviations for random matrices and random graphs. Unpublished manuscript.
[9] Boucheron, S., Lugosi, G. and Massart, P. (2003). Concentration inequalities using the entropy method. Ann. Probab. 31 1583–1614.
Mathematical Reviews (MathSciNet): MR1989444
Zentralblatt MATH: 1051.60020
Digital Object Identifier: doi:10.1214/aop/1055425791
Project Euclid: euclid.aop/1055425791
[10] Chatterjee, S. (2005). Concentration inequalities with exchangeable pairs. Ph.D. thesis, Stanford Univ. Available at arXiv:math/0507526.
Mathematical Reviews (MathSciNet): MR2707160
[11] Chatterjee, S. (2007). Stein’s method for concentration inequalities. Probab. Theory Related Fields 138 305–321.
Mathematical Reviews (MathSciNet): MR2288072
Zentralblatt MATH: 1116.60056
Digital Object Identifier: doi:10.1007/s00440-006-0029-y
[12] Chatterjee, S. (2007). Concentration of Haar measures, with an application to random matrices. J. Funct. Anal. 245 379–389.
Mathematical Reviews (MathSciNet): MR2309833
Zentralblatt MATH: 1115.60007
Digital Object Identifier: doi:10.1016/j.jfa.2007.01.003
[13] Chatterjee, S. and Shao, Q.-M. (2009). Stein’s method of exchangeable pairs with application to the Curie–Weiss model. Preprint. Available at arXiv:0907.4450.
[14] Chazottes, J. R., Collet, P., Külske, C. and Redig, F. (2007). Concentration inequalities for random fields via coupling. Probab. Theory Related Fields 137 201–225.
[15] Döring, H. and Eichelsbacher, P. (2009). Moderate deviations in a random graph and for the spectrum of Bernoulli random matrices. Electron. J. Probab. 14 2636–2656.
Mathematical Reviews (MathSciNet): MR2570014
Zentralblatt MATH: 1193.60032
[16] Eichelsbacher, P. and Lowe, M. (2009). Stein’s method for dependent random variables occurring in statistical mechanics. Preprint. Available at arXiv:0908.1909.
[17] Ellis, R. S. and Newman, C. M. (1978). The statistics of Curie–Weiss models. J. Stat. Phys. 19 149–161.
Mathematical Reviews (MathSciNet): MR503332
Digital Object Identifier: doi:10.1007/BF01012508
[18] Ellis, R. S. and Newman, C. M. (1978). Limit theorems for sums of dependent random variables occurring in statistical mechanics. Z. Wahrsch. Verw. Gebiete 44 117–139.
Mathematical Reviews (MathSciNet): MR503333
[19] Ellis, R. S. (1985). Entropy, Large Deviations, and Statistical Mechanics. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 271. Springer, New York.
Mathematical Reviews (MathSciNet): MR793553
[20] Gentil, I., Guillin, A. and Miclo, L. (2005). Modified logarithmic Sobolev inequalities and transportation inequalities. Probab. Theory Related Fields 133 409–436.
Mathematical Reviews (MathSciNet): MR2198019
Zentralblatt MATH: 1080.26010
Digital Object Identifier: doi:10.1007/s00440-005-0432-9
[21] Gozlan, N. (2007). Characterization of Talagrand’s like transportation-cost inequalities on the real line. J. Funct. Anal. 250 400–425.
Mathematical Reviews (MathSciNet): MR2352486
Zentralblatt MATH: 1135.46022
Digital Object Identifier: doi:10.1016/j.jfa.2007.05.025
[22] Gozlan, N. (2010). Poincare inequalities and dimension free concentration of measure. Ann. Inst. H. Poincaré Probab. Statist. To appear.
[23] Ising, E. (1925). Beitrag zur theorie des ferromagnetismus. Zeitschrift für Physik A Hadrons and Nuclei 31 253–258.
[24] Janson, S., Łuczak, T. and Rucinski, A. (2000). Random Graphs. Wiley, New York.
Mathematical Reviews (MathSciNet): MR1782847
[25] Janson, S., Oleszkiewicz, K. and Ruciński, A. (2004). Upper tails for subgraph counts in random graphs. Israel J. Math. 142 61–92.
Mathematical Reviews (MathSciNet): MR2085711
Zentralblatt MATH: 1055.05136
Digital Object Identifier: doi:10.1007/BF02771528
[26] Janson, S. and Ruciński, A. (2002). The infamous upper tail: Probabilistic methods in combinatorial optimization. Random Structures Algorithms 20 317–342.
Mathematical Reviews (MathSciNet): MR1900611
[27] Kim, J. H. and Vu, V. H. (2004). Divide and conquer martingales and the number of triangles in a random graph. Random Structures Algorithms 24 166–174.
Mathematical Reviews (MathSciNet): MR2035874
[28] Latała, R. and Oleszkiewicz, K. (2000). Between Sobolev and Poincaré. In Geometric Aspects of Functional Analysis. Lecture Notes in Math. 1745 147–168. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR1796718
[29] Ledoux, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. Amer. Math. Soc., Providence, RI.
Mathematical Reviews (MathSciNet): MR1849347
[30] Martin-Löf, A. (1982). A Laplace approximation for sums of independent random variables. Z. Wahrsch. Verw. Gebiete 59 101–115.
Mathematical Reviews (MathSciNet): MR643791
[31] Onsager, L. (1944). Crystal statistics. I. A two-dimensional model with an order–disorder transition. Phys. Rev. (2) 65 117–149.
Mathematical Reviews (MathSciNet): MR10315
Digital Object Identifier: doi:10.1103/PhysRev.65.117
[32] Park, J. and Newman, M. E. J. (2004). Statistical mechanics of networks. Phys. Rev. E (3) 70 066117–066122.
Mathematical Reviews (MathSciNet): MR2133807
[33] Park, J. and Newman, M. E. J. (2005). Solution for the properties of a clustered network. Phys. Rev. E 72 026136–026137.
[34] Raič, M. (2007). CLT-related large deviation bounds based on Stein’s method. Adv. in Appl. Probab. 39 731–752.
Mathematical Reviews (MathSciNet): MR2357379
Zentralblatt MATH: 1127.60024
Digital Object Identifier: doi:10.1239/aap/1189518636
Project Euclid: euclid.aap/1189518636
[35] Simon, B. and Griffiths, R. B. (1973). The (ϕ4)2 field theory as a classical Ising model. Comm. Math. Phys. 33 145–164.
Mathematical Reviews (MathSciNet): MR428998
Digital Object Identifier: doi:10.1007/BF01645626
Project Euclid: euclid.cmp/1103859251
[36] Stein, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proc. Sixth Berkeley Symp. Math. Statist. Probab. Vol. II: Probability Theory 583–602. Univ. California Press, Berkeley, CA.
Mathematical Reviews (MathSciNet): MR402873
Zentralblatt MATH: 0278.60026
[37] Stein, C. (1986). Approximate Computation of Expectations. IMS, Hayward, CA.
Mathematical Reviews (MathSciNet): MR882007
Zentralblatt MATH: 0721.60016
[38] Talagrand, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Publ. Math. Inst. Hautes Études Sci. 81 73–205.
Mathematical Reviews (MathSciNet): MR1361756
Digital Object Identifier: doi:10.1007/BF02699376
[39] Vu, V. H. (2001). A large deviation result on the number of small subgraphs of a random graph. Combin. Probab. Comput. 10 79–94.
Mathematical Reviews (MathSciNet): MR1827810
Zentralblatt MATH: 0982.05092
previous :: next
