Stein's method in high dimensions with applications
Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 2, p. 529-549

Let h be a three times partially differentiable function on n , let X=(X 1 ,...,X n ) be a collection of real-valued random variables and let Z=(Z 1 ,...,Z n ) be a multivariate Gaussian vector. In this article, we develop Stein’s method to give error bounds on the difference 𝔼h(X)-𝔼h(Z) in cases where the coordinates of X are not necessarily independent, focusing on the high dimensional case n. In order to express the dependency structure we use Stein couplings, which allows for a broad range of applications, such as classic occupancy, local dependence, Curie-Weiss model, etc. We will also give applications to the Sherrington-Kirkpatrick model and last passage percolation on thin rectangles.

Soit h une fonction réelle sur n dont les dérivées partielles d’ordre trois existent, soit X=(X 1 ,...,X n ) un vecteur de variables aléatoire réelles et soit Z=(Z 1 ,...,Z n ) un vecteur aléatoire Gaussien. Dans cet article, nous établissons par la méthode de Stein une majoration de la différence 𝔼h(X)-𝔼h(Z) dans le cas où les coordonnées de X ne sont pas nécessairement indépendantes; nous nous concentrons sur le cas de la grande dimension n. Pour exprimer la structure de dépendance, nous utilisons des couplages de Stein, ce qui permet une large gamme d’applications, par exemple aux modèles d’urnes, au modèles avec dépendance locale, au modèle de Curie-Weiss, etc. Nous présentons aussi des applications au modèle de Sherrington-Kirkpatrick et à la percolation de dernier passage dans des rectangles étroits.

DOI : https://doi.org/10.1214/11-AIHP473
Classification:  60F17,  82B44
Keywords: Stein's method, gaussian interpolation, last passage percolation on thin rectangles, Sherrington-Kirkpatrick model, Curie-Weiss model
@article{AIHPB_2013__49_2_529_0,
     author = {R\"ollin, Adrian},
     title = {Stein's method in high dimensions with applications},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {49},
     number = {2},
     year = {2013},
     pages = {529-549},
     doi = {10.1214/11-AIHP473},
     zbl = {1287.60043},
     mrnumber = {3088380},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2013__49_2_529_0}
}
Röllin, Adrian. Stein's method in high dimensions with applications. Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 2, pp. 529-549. doi : 10.1214/11-AIHP473. http://www.numdam.org/item/AIHPB_2013__49_2_529_0/

[1] J. Baik and T. M. Suidan. A GUE central limit theorem and universality of directed first and last passage site percolation. Int. Math. Res. Not. 2005 (2005) 325-337. | MR 2131383 | Zbl 1136.60313

[2] A. D. Barbour, M. Karoński and A. Ruciński. A central limit theorem for decomposable random variables with applications to random graphs. J. Combin. Theory Ser. B 47 (1989) 125-145. | MR 1047781 | Zbl 0689.05042

[3] T. Bodineau and J. Martin. A universality property for last-passage percolation paths close to the axis. Electron. Commun. Probab. 10 (2005) 105-112 (electronic). | MR 2150699 | Zbl 1111.60068

[4] E. Bolthausen. Exact convergence rates in some martingale central limit theorems. Ann. Probab. 10 (1982) 672-688. | MR 659537 | Zbl 0494.60020

[5] P. Carmona and Y. Hu. Universality in Sherrington-Kirkpatrick's spin glass model. Ann. Inst. Henri Poincaré Probab. Stat. 42 (2006) 215-222. | Numdam | MR 2199799 | Zbl 1099.82005

[6] S. Chatterjee. A simple invariance theorem. Preprint, 2005. Available at http://arxiv.org/abs/math.PR/0508213.

[7] S. Chatterjee. A generalization of the Lindeberg principle. Ann. Probab. 34 (2006) 2061-2076. | MR 2294976 | Zbl 1117.60034

[8] S. Chatterjee and E. Meckes. Multivariate normal approximation using exchangeable pairs. ALEA Lat. Am. J. Probab. Math. Stat. 4 (2008) 257-283. | MR 2453473 | Zbl 1162.60310

[9] S. Chatterjee and Q.-M. Shao. Non-normal approximation by Stein's method of exchangeable pairs with application to the Curie-Weiss model. Ann. Appl. Probab. 21 (2011) 464-483. | MR 2807964 | Zbl 1216.60018

[10] L. H. Y. Chen and A. Röllin. Stein couplings for normal approximation. Preprint, 2010. Available at http://arxiv.org/abs/1003.6039.

[11] A. Dembo and Y. Rinott. Some examples of normal approximations by Stein's method. In Random Discrete Structures (Minneapolis, MN, 1993) 25-44. IMA Vol. Math. Appl. 76. Springer, New York, 1996. | MR 1395606 | Zbl 0847.60015

[12] P. Eichelsbacher and M. Löwe. Stein's method for dependent random variables occurring in statistical mechanics. Electron. J. Probab. 15 (2010) 962-988. | MR 2659754 | Zbl 1225.60042

[13] L. Erdős, H.-T. Yau and J. Yin. Bulk universality for generalized Wigner matrices. Preprint, 2010. Available at arxiv.org/abs/1001.3453. | MR 2981427 | Zbl 1277.15026

[14] F. Götze and A. N. Tikhomirov. Limit theorems for spectra of random matrices with martingale structure. Teor. Veroyatn. Primen. 51 (2006) 171-192. | MR 2324173 | Zbl 1118.15022

[15] I. G. Grama. On moderate deviations for martingales. Ann. Probab. 25 (1997) 152-183. | MR 1428504 | Zbl 0881.60026

[16] K. Johansson. Shape fluctuations and random matrices. Comm. Math. Phys. 209 (2000) 437-476. | MR 1737991 | Zbl 0969.15008

[17] K. Johansson. Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Comm. Math. Phys. 215 (2001) 683-705. | MR 1810949 | Zbl 0978.15020

[18] E. S. Meckes. On Stein's method for multivariate normal approximation. In High Dimensional Probability V: The Luminy Volume 153-178. Inst. Math. Statist., Beachwood, OH, 2009. | MR 2797946 | Zbl 1243.60025

[19] E. Mossel, R. O'Donnell and K. Oleszkiewicz. Noise stability of functions with low influences: Invariance and optimality. Ann. of Math. 171 (2010) 295-341. | MR 2630040 | Zbl 1201.60031

[20] M. Raič. A multivariate CLT for decomposable random vectors with finite second moments. J. Theoret. Probab. 17 (2004) 573-603. | MR 2091552 | Zbl 1059.62050

[21] G. Reinert and A. Röllin. Multivariate normal approximation with Stein's method of exchangeable pairs under a general linearity condition. Ann. Probab. 37 (2009) 2150-2173. | MR 2573554 | Zbl 1200.62010

[22] I. Rinott and V. I. Rotar'. Some estimates for the rate of convergence in the CLT for martingales. II. Theory Probab. Appl. 44 (1999) 523-536. | MR 1805821 | Zbl 0969.60036

[23] Y. Rinott and V. Rotar'. A multivariate CLT for local dependence with n -1/2 logn rate and applications to multivariate graph related statistics. J. Multivariate Anal. 56 (1996) 333-350. | MR 1379533 | Zbl 0859.60019

[24] V. I. Rotar'. Certain limit theorems for polynomials of degree two. Teor. Veroyatn. Primen. 18 (1973) 527-534. Actual title is “Some limit theorems for polynomials of degree two”. | MR 326803 | Zbl 0304.60037

[25] D. Slepian. The one-sided barrier problem for Gaussian noise. Bell System Tech. J. 41 (1962) 463-501. | MR 133183

[26] C. Stein. 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, 1972. | MR 402873 | Zbl 0278.60026

[27] T. Suidan. A remark on a theorem of Chatterjee and last passage percolation. J. Phys. A 39 (2006) 8977-8981. | MR 2240468 | Zbl 1148.82014

[28] M. Talagrand. The Parisi formula. Ann. of Math. 163 (2006) 221-263. | MR 2195134 | Zbl 1137.82010

[29] M. Talagrand. Mean Field Models for Spin Glasses. Volume I. Springer-Verlag, Berlin, 2010. | MR 2731561 | Zbl 1214.82002

[30] T. Tao and V. Vu. Random matrices: universality of local eigenvalue statistics. Acta Math. 206 (2011) 127-204. | MR 2784665 | Zbl 1217.15043