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Concentration of empirical distribution functions with applications to non-i.i.d. models. (English) Zbl 1207.62106

Summary: The concentration of empirical measures is studied for dependent data, whose joint distribution satisfies Poincaré-type or logarithmic Sobolev inequalities. The general concentration results are then applied to spectral empirical distribution functions associated with high-dimensional random matrices.

MSC:

62G30 Order statistics; empirical distribution functions
62H10 Multivariate distribution of statistics
60E15 Inequalities; stochastic orderings
62E17 Approximations to statistical distributions (nonasymptotic)
60F10 Large deviations
15B52 Random matrices (algebraic aspects)
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[1] Ball, K. (1988). Logarithmically concave functions and sections of convex sets in R n . Studia Math. 88 69-84. · Zbl 0642.52011
[2] Bhatria, R. (1997). Matrix Analysis. Graduate Texts in Mathematics 169 . New York: Springer.
[3] Bobkov, S.G. (1999). Remarks on the Gromov-Milman inequality. Vestn. Syktyvkar. Univ. Ser. 1 Mat. Mekh. Inform. 3 15-22. · Zbl 0981.60008
[4] Bobkov, S.G. (1999). Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab. 27 1903-1921. · Zbl 0964.60013
[5] Bobkov, S.G., Gentil, I. and Ledoux, M. (2001). Hypercontractivity of Hamilton-Jacobi equations. J. Math. Pures Appl. 88 669-696. · Zbl 1038.35020
[6] Bobkov, S.G. and Götze, F. (1999). Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 1 1-28. · Zbl 0924.46027
[7] Bobkov, S.G. and Götze, F. (2008). Hardy-type inequalities via Riccati and Sturm-Liouville equations. In Sobolev Spaces in Mathematics, I (V. Maz’ya, ed.). Intern. Math. Series 8 69-86. New York: Springer. · Zbl 1173.26319
[8] Bobkov, S.G., Götze, F. and Tikhomirov, A.N. (2010). On concentration of empirical measures and convergence to the semi-circle law. Bielefeld University. Preprint. J. Theor. Probab. · Zbl 1247.60010
[9] Bobkov, S.G. and Houdré, C. (1997). Isoperimetric constants for product probability measures. Ann. Probab. 25 184-205. · Zbl 0878.60013
[10] Bobkov, S.G. and Ledoux, M. (1997). Poincare’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution. Probab. Theory Related Fields 107 383-400. · Zbl 0878.60014
[11] Bobkov, S.G. and Ledoux, M. (2009). Weighted Poincaré-type inequalities for Cauchy and other convex measures. Ann. Probab. 37 403-427. · Zbl 1178.46041
[12] Borovkov, A.A. and Utev, S.A. (1983). On an inequality and a characterization of the normal distribution connected with it. Probab. Theory Appl. 28 209-218. · Zbl 0511.60016
[13] Cambanis, S., Simons, G. and Stout, W. (1976). Inequalities for Ek ( X , Y ) when the marginals are fixed. Z. Wahrsch. Verw. Gebiete 36 285-294. · Zbl 0325.60002
[14] Chatterjee, S. and Bose, A. (2004). A new method for bounding rates of convergence of empirical spectral distributions. J. Theoret. Probab. 17 1003-1019. · Zbl 1063.60024
[15] Davidson, K.R. and Szarek, S.J. (2001). Local operator theory, random matrices and Banach spaces. In Handbook of the Geometry of Banach Spaces I 317-366. Amsterdam: North-Holland. · Zbl 1067.46008
[16] Dudley, R.M. (1989). Real Analysis and Probability . Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software. · Zbl 0686.60001
[17] Dvoretzky, A., Kiefer, J. and Wolfowitz, J. (1956). Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator. Ann. Math. Statist. 27 642-669. · Zbl 0073.14603
[18] Erdös, L., Schlein, B. and Yau, H.-T. (2009). Local semicircle law and complete delocalization for Wigner random matrices. Comm. Math. Phys. 287 641-655. · Zbl 1186.60005
[19] Evans, L.C. (1997). Partial Differential Equations. Graduate Studies in Math. 19 . Providence, RI: Amer. Math. Soc. · Zbl 0980.65501
[20] Götze, F. and Tikhomirov, A.N. (2003). Rate of convergence to the semi-circular law. Probab. Theory Related Fields 127 228-276. · Zbl 1031.60019
[21] Götze, F. and Tikhomirov, A.N. (2005). The rate of convergence for spectra of GUE and LUE matrix ensembles. Cent. Eur. J. Math. 3 666-704 (electronic). · Zbl 1108.60014
[22] Götze, F., Tikhomirov, A.N. and Timushev, D.A. (2007). Rate of convergence to the semi-circle law for the deformed Gaussian unitary ensemble. Cent. Eur. J. Math. 5 305-334 (electronic). · Zbl 1155.15027
[23] Gromov, M. and Milman, V.D. (1983). A topological application of the isoperimetric inequality. Amer. J. Math. 105 843-854. · Zbl 0522.53039
[24] Guionnet, A. and Zeitouni, O. (2000). Concentration of the spectral measure for large matrices. Electron. Comm. Probab. 5 119-136. · Zbl 0969.15010
[25] Gustavsson, J. (2005). Gaussian fluctuations of eigenvalues in the GUE. Ann. Inst. H. Poincaré Probab. Statist. 41 151-178. · Zbl 1073.60020
[26] Hensley, D. (1980). Slicing convex bodies - bounds for slice area in terms of the body’s covariance. Proc. Amer. Math. Soc. 79 619-625. · Zbl 0439.52008
[27] Kac, I.S. and Krein, M.G. (1958). Criteria for the discreteness of the spectrum of a singular string. Izv. Vysš. Učebn. Zaved. Matematika 2 136-153.
[28] Kim, T.Y. (1999). On tail probabilities of Kolmogorov-Smirnov statistics based on uniform mixing processes. Statist. Probab. Lett. 43 217-223. · Zbl 0929.62010
[29] Ledoux, M. (1999). Concentration of measure and logarithmic Sobolev inequalities. In Séminaire de Probabilités XXXIII. Lecture Notes in Math. 1709 120-216. Berlin: Springer. · Zbl 0957.60016
[30] Ledoux, M. (2001). The Concentration of Measure Phenomenon. Math. Surveys and Monographs 89 . Providence, RI: Amer. Math. Soc. · Zbl 0995.60002
[31] Ledoux, M. (2007). Deviation inequalities on largest eigenvalues. In Geom. Aspects of Funct. Anal., Israel Seminar 2004-2005. Lecture Notes in Math. 1910 167-219. Berlin: Springer. · Zbl 1130.15012
[32] Massart, P. (1990). The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality. Ann. Probab. 18 1269-1283. · Zbl 0713.62021
[33] Maurey, B. (1991). Some deviation inequalities. Geom. Funct. Anal. 1 188-197. · Zbl 0756.60018
[34] Maz’ya, V.G. (1985). Sobolev Spaces . Berlin: Springer.
[35] Muckenhoupt, B. (1972). Hardy’s inequality with weights. Studia Math. XLIV 31-38. · Zbl 0236.26015
[36] Pastur, L.A. (1973). Spectra of random selfadjoint operators. Uspehi Mat. Nauk 28 3-64.
[37] Ruschendorf, L. (1985). The Wasserstein distance and approximation theorems. Z. Wahrsch. Verw. Gebiete 70 117-129. · Zbl 0554.60024
[38] Sen, P.K. (1974). Weak convergence of multidimensional empirical processes for stationary \varphi -mixing processes. Ann. Probab. 2 147-154. · Zbl 0276.60030
[39] Soshnikov, A. (1999). Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. 207 697-733. · Zbl 1062.82502
[40] Talagrand, M. (1996). Majorizing measures: The generic chaining. Ann. Probab. 24 1049-1103. · Zbl 0867.60017
[41] Tao, T. and Vu, V. (2009). Random matrices: Universality of local eigenvalue statistics. · Zbl 1217.15043
[42] Timushev, D.A. (2006). On the rate of convergence in probability of the spectral distribution function of a random matrix. Teor. Veroyatn. Primen. 51 618-622. · Zbl 1128.60019
[43] Vallander, S.S. (1973). Calculations of the Vasserstein distance between probability distributions on the line. Teor. Verojatn. Primen. 18 824-827. · Zbl 0351.60009
[44] Yoshihara, K. (1975/76). Weak convergence of multidimensional empirical processes for strong mixing sequences of stochastic vectors. Z. Wahrsch. Verw. Gebiete 33 133-137. · Zbl 0304.60019
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