Exact rates in Vapnik-Chervonenkis bounds
Vayatis, Nicolas
Annales de l'I.H.P. Probabilités et statistiques, Tome 39 (2003), p. 95-119 / Harvested from Numdam
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     author = {Vayatis, Nicolas},
     title = {Exact rates in Vapnik-Chervonenkis bounds},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {39},
     year = {2003},
     pages = {95-119},
     zbl = {1020.60010},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2003__39_1_95_0}
}

Vayatis, Nicolas. Exact rates in Vapnik-Chervonenkis bounds. Annales de l'I.H.P. Probabilités et statistiques, Tome 39 (2003) pp. 95-119. http://gdmltest.u-ga.fr/item/AIHPB_2003__39_1_95_0/

[1] K. Alexander, Probability inequalities for empirical processes and a law of the iterated logarithm, Ann. Probab. 4 (1984) 1041-1067. | MR 757769 | Zbl 0549.60024

[2] R. Azencott, Communication for the fifty years of the Department of Mathematics at Brown University (USA), 1996.

[3] S. Boucheron, G. Lugosi, P. Massart, A sharp concentration inequality with applications, Random Structures and Algorithms 16 (3) (2000) 277-292. | MR 1749290 | Zbl 0954.60008

[4] A. Dembo, O. Zeitouni, Large Deviations Techniques and Applications, Springer, 1998. | MR 1619036 | Zbl 0896.60013

[5] L. Devroye, Bounds for the uniform deviation of empirical measures, J. Multivariate Anal. 12 (1982) 72-79. | MR 650929 | Zbl 0492.60006

[6] L. Devroye, L. Györfi, G. Lugosi, A Probabilistic Theory of Pattern Recognition, Springer, 1996. | MR 1383093 | Zbl 0853.68150

[7] R.M. Dudley, A course on empirical processes, in: Hennequin P.L. (Ed.), Ecole d'Eté de Probabilités de Saint-Flour XII - 1982, Lecture Notes in Mathematics, 1097, Springer-Verlag, 1982, pp. 1-142. | MR 876079 | Zbl 0554.60029

[8] A. Dvoretzky, J. Kiefer, J. Wolfowitz, Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator, Ann. Math. Statist. 27 (1956) 642-669. | MR 83864 | Zbl 0073.14603

[9] E. Giné, J. Zinn, On the central limit theorem for empirical processes, Ann. Probab. 12 (1984) 929-989. | MR 757767 | Zbl 0553.60037

[10] D. Haussler, Sphere packing numbers for subsets of the Boolean n-cube with bounded Vapnik-Chervonenkis dimension, J. Combin. Theory Series A 69 (1995) 217-232. | Zbl 0818.60005

[11] W. Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58 (1963) 13-30. | MR 144363 | Zbl 0127.10602

[12] J. Kiefer, On large deviations of the empirical d.f. of vector chance variables and a law of the iterated logarithm, Pacific J. Math. 11 (1961) 649-660. | MR 131885 | Zbl 0119.34904

[13] M. Ledoux, M. Talagrand, Probability in Banach Spaces, Springer-Verlag, 1992. | MR 1102015 | Zbl 0748.60004

[14] G. Lugosi, Improved upper bounds for probabilities of uniform deviations, Statist. Probab. Lett. 25 (1995) 71-77. | MR 1364820 | Zbl 0839.60020

[15] P. Massart, Rates of convergence in the central limit theorem for empirical processes, Annales de l'Institut Henri Poincaré 22 (4) (1986) 381-423. | Numdam | MR 871904 | Zbl 0615.60032

[16] P. Massart, The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality, Ann. Probab. 18 (1990) 1269-1283. | Zbl 0713.62021

[17] J.M.R. Parrondo, C. Van Den Broeck, Vapnik-Chervonenkis bounds for generalization, J. Phys. A: Math. Gen. 26 (1993) 2211-2223. | Zbl 0777.60098

[18] D. Pollard, Convergence of Stochastic Processes, Springer-Verlag, 1984. | MR 762984 | Zbl 0544.60045

[19] D. Pollard, Empirical Processes: Theory and Applications, NSF-CBMS Regional Conference Series in Probability and Statistics, 2, Institute of Mathematical Statistics, 1991. | MR 1089429 | Zbl 0741.60001

[20] E. Rio, Inégalités de concentration pour les processus empiriques de classes de parties, Probab. Theory Related Fields (2000), to appear. | MR 1818244 | Zbl 0976.60033

[21] M. Talagrand, Sharper bounds for Gaussian and empirical processes, Ann. Probab. 22 (1) (1994) 28-76. | MR 1258865 | Zbl 0798.60051

[22] A.W. Van Der Vaart, J.A. Wellner, Weak Convergence and Empirical Processes, Springer, 1996. | MR 1385671 | Zbl 0862.60002

[23] V.N. Vapnik, Estimation of Dependencies on the Basis of Empirical Data, Springer, 1982.

[24] V.N. Vapnik, The Nature of Statistical Learning Theory, Springer, 1995. | MR 1367965 | Zbl 0833.62008

[25] V.N. Vapnik, Statistical Learning Theory, Wiley-Interscience, 1998. | MR 1641250 | Zbl 0935.62007

[26] V.N. Vapnik, A.Y. Chervonenkis, On the uniform convergence of relative frequencies of events to their probabilities, Theory Probab. Appl. 16 (1971) 264-280. | Zbl 0247.60005

[27] V.N. Vapnik, A.Y. Chervonenkis, Necessary and sufficient conditions for the uniform convergence of the means to their expectations, Theory Probab. Appl. 26 (1981) 532-555. | MR 627861 | Zbl 0487.60036

[28] V.N. Vapnik, E. Levin, Y. Le Cun, Measuring the VC-dimension of a learning machine, Neural Comput. 6 (1994) 851-876.

[29] N. Vayatis, Inégalités de Vapnik-Chervonenkis et mesures de complexité, Ph.D. thesis, Ecole Polytechnique, 2000, in English.

[30] L. Wu, Large deviations, moderate deviations and LIL for empirical processes, Ann. Probab. 22 (1) (1994) 17-27. | MR 1258864 | Zbl 0793.60032