Abstract
Let F be a class of functions on a probability space (Ω, μ) and let X 1,...,X k be independent random variables distributed according to μ. We establish an upper bound that holds with high probability on \({\rm sup}_{f \in F} |\{i : |f(X_i)| \geq t \}\) for every t > 0, and that depends on a natural geometric parameter associated with F. We use this result to analyze the supremum of empirical processes of the form \(Z_f = \left|k^{-1}\sum_{i=1}^k |f|^p(X_i) - {\mathbb{E}}|f|^p\right|\) for p > 1 using the geometry of F. We also present some geometric applications of this approach, based on properties of the random operator \(\Gamma = k^{-1/2}\sum_{i=1}^k\) 〈X i , ·〉e i , where \((X_i)_{i=1}^k\) are sampled according to an isotropic, log-concave measure on \({\mathbb{R}}^n\) .
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References
Barthe F., Guédon O., Mendelson S. and Naor A. (2005). A probabilistic approach to the geometry of the \(\ell_p^n\) ball. Ann. Probab. 33(2): 480–513
Borell C. (1975). The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30: 207–216
Bourgain, J.: Random points in isotropic convex bodies. Convex Geometric Analysis (Berkeley, CA, 1996) Math. Sci. Res. Inst. Publ. 34, 53–58 (1999)
Giannopoulos A.A. and Milman V.D. (2000). Concentration property on probability spaces. Adv. Math. 156: 77–106
Giné E. and Zinn J. (1984). Some limit theorems for empirical processes. Ann. Probab. 12(4): 929–989
Gordon Y., Litvak A., Schütt C. and Werner E. (2002). Orlicz norms of sequences of random varialbes. Ann. Probab. 30: 1833–1853
Guédon O. and Rudelson M. (2007). L p moments of random vectors via majorizing measures. Adv. Math. 208(2): 798–823
Kannan R., Lovász L. and Simonovits M. (1997). Random walks and O *(n 5) volume algorithm for convex bodies. Random Struct. Algorithms 2(1): 1–50
Ledoux, M.: The concentration of measure phenomenon. Mathematical Surveys an Monographs, vol. 89, AMS (2001)
Ledoux, M., Talagrand, M.: Probability in Banach spaces. Isoperimetry and processes. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 23. Springer, Berlin (1991)
Lust-Piquard F. and Pisier G. (1991). Non-commutative Khinchine and Paley inequalities. Ark. Mat. 29: 241–260
Mendelson, S., Pajor, A., Tomczak-Jaegermann, N.: Reconstruction and subgaussian operators. Geom. Funct. Anal. (to appear)
Milman, V.D., Schechtman, G.: Asymptotic theory of finite dimensional normed spaces. Lecture Notes in Mathematics 1200. Springer, Heidelberg (1986)
Pajor A. and Tomczak-Jaegermann N. (1986). Subspaces of small codimension of finite-dimensional Banach spaces. Proc. AMS 97(4): 637–642
Pajor A. and Tomczak-Jaegermann N. (1985). Remarques sur les nombres d’entropie d’umopérteur et de son transposé. C. R. Acad. Sci. Paris 301: 743–746
Paouris G. (2006). Concentration of mass on convex bodies. Geom. Funct. Anal. 16(5): 1021–1049
Pisier G. (1989). The Volume of Convex Bodies and Banach Space Geometry. Cambridge University Press, Cambridge
Rudelson M. (1999). Random vectors in the isotropic position. J. Funct. Anal. 164: 60–72
Talagrand M. (1987). Regularity of gaussian processes. Acta Math. 159: 99–149
Talagrand M. (1994). Sharper bounds for Gaussian and empirical processes. Ann. Probab. 22(1): 28–76
Talagrand M. (1996). Majorizing measures: the generic chaining. Ann. Probab. 24(3): 1049–1103
Talagrand M. (2005). The Generic Chaining. Springer, Heidelberg
Van der Vaart A.W. and Wellner J.A. (1996). Weak convergence and empirical processes. Springer, Heidelberg
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Mendelson, S. On weakly bounded empirical processes. Math. Ann. 340, 293–314 (2008). https://doi.org/10.1007/s00208-007-0152-9
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DOI: https://doi.org/10.1007/s00208-007-0152-9