On weakly bounded empirical processes

Let F be a class of functions on a probability space (Ω, μ) and let X1,...,Xk 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 $$\Gamma = k^{-1/2}\sum_{i=1}^k$$ 〈Xi, ·〉ei, where $$(X_i)_{i=1}^k$$ are sampled according to an isotropic, log-concave measure on $${\mathbb{R}}^n$$ .

[1]  C. Borell The Brunn-Minkowski inequality in Gauss space , 1975 .

[2]  E. Giné,et al.  Some Limit Theorems for Empirical Processes , 1984 .

[3]  A. Pajor,et al.  Subspaces of small codimension of finite-dimensional Banach spaces , 1986 .

[4]  Gilles Pisier,et al.  Banach spaces with a weak cotype 2 property , 1986 .

[5]  V. Milman,et al.  Asymptotic Theory Of Finite Dimensional Normed Spaces , 1986 .

[6]  M. Talagrand Regularity of gaussian processes , 1987 .

[7]  G. Pisier The volume of convex bodies and Banach space geometry , 1989 .

[8]  Miklós Simonovits,et al.  Szemerédi's Partition and Quasirandomness , 1991, Random Struct. Algorithms.

[9]  M. Talagrand,et al.  Probability in Banach Spaces: Isoperimetry and Processes , 1991 .

[10]  M. Talagrand,et al.  Probability in Banach spaces , 1991 .

[11]  G. Pisier,et al.  Non commutative Khintchine and Paley inequalities , 1991 .

[12]  M. Talagrand Sharper Bounds for Gaussian and Empirical Processes , 1994 .

[13]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

[14]  M. Talagrand Majorizing measures: the generic chaining , 1996 .

[15]  M. Rudelson Random Vectors in the Isotropic Position , 1996, math/9608208.

[16]  M. Simonovits,et al.  Random walks and an O * ( n 5 ) volume algorithm for convex bodies , 1997 .

[17]  P. Gänssler Weak Convergence and Empirical Processes - A. W. van der Vaart; J. A. Wellner. , 1997 .

[18]  V. Milman,et al.  Concentration Property on Probability Spaces , 2000 .

[19]  M. Ledoux The concentration of measure phenomenon , 2001 .

[20]  Alexander E. Litvak,et al.  Orlicz norms of sequences of random variables , 2002 .

[21]  V. Milman,et al.  Regularization of star bodies by random hyperplane cut off , 2003 .

[22]  M. Talagrand The Generic Chaining , 2005 .

[23]  Shahar Mendelson,et al.  The Geometry of Random {-1,1}-Polytopes , 2005, Discret. Comput. Geom..

[24]  S. Mendelson,et al.  A probabilistic approach to the geometry of the ℓᵨⁿ-ball , 2005, math/0503650.

[25]  A. Giannopoulos,et al.  Random Points in Isotropic Unconditional Convex Bodies , 2005 .

[26]  M. Rudelson,et al.  Lp-moments of random vectors via majorizing measures , 2005, math/0507023.

[27]  S. Mendelson,et al.  Reconstruction and subgaussian operators , 2005, math/0506239.

[28]  G. Paouris Concentration of mass on convex bodies , 2006 .

[29]  S. Mendelson,et al.  Reconstruction and Subgaussian Operators in Asymptotic Geometric Analysis , 2007 .