Stein's method in high dimensions with applications

Röllin, Adrian

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 2, pp. 529-549.

Soit $h$ une fonction réelle sur ${ℝ}^n$ 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 $𝔼h\left(X\right)-𝔼h\left(Z\right)$ dans le cas où les coordonnées de $X$ ne sont pas nécessairement indépendantes; nous nous concentrons sur le cas de la grande dimension $n\to \infty$. 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.

Let $h$ be a three times partially differentiable function on ${ℝ}^n$, 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 $𝔼h\left(X\right)-𝔼h\left(Z\right)$ in cases where the coordinates of $X$ are not necessarily independent, focusing on the high dimensional case $n\to \infty$. 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.

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

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

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

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