On measure concentration of vector-valued maps

• Autorzy: Ledoux M. , Oleszkiewicz K.
• Języki publikacji: EN

Abstrakty

EN

We study concentration properties for vector-valued maps. In particular, we describe inequalities which capture the exact dimensional behavior of Lipschitz maps with values in R^k. To this end, we study in particular a domination principle for projections which might be of independent interest. We further compare our conclusions with earlier results by Pinelis in the Gaussian case, and discuss extensions to the infinite-dimensional setting.

Słowa kluczowe

EN

concentration of measure; vector-valued map; moment comparison; Gaussian measure

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2007
• Tom: Vol. 55, no 3
• Strony: 261--278
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Ledoux M.

autor

Oleszkiewicz K.

• Institut de Mathematiques, Universite Paul-Sabatier, 31062 Toulouse, France,

Bibliografia

• [1] F. Barthe, Extremal properties of central half-spaces for product measures, J. Funct. Anal. 182 (2001), 81-107.
• [2] Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 (1991), 375-417.
• [3] L. Caffarelli, Monotonicity properties of optimal transportation and the FKG and related inequalities, Comm. Math. Phys. 214 (2000), 547-563.
• [4] M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Birkhauser, 1998.
• [5] M. Gromov, Isoperimetry of waists and concentration of maps, Geom. Funct. Anal. 13 (2003), 178-215.
• [6] S. Kwapień and W. A. Woyczyński, Random Series and Stochastic Integrals: Single and Multiple, Probab. Appl., Birkhauser, 1992.
• [7] R. Latała and K. Oleszkiewicz, Small ball probability estimates in terms of width, Studia Math. 169 (2005), 305-314.
• [8] M. Ledoux, The Concentration of Measure Phenomenon, Math. Surveys Monogr. 89, Amer. Math. Soc., 2001.
• [9] M. Ledoux and M. Talagrand, Probability in Banach Spaces. Isoperimetry and Processes, Ergeb. Math. Grenzgeb. (3) 23, Springer, 1991.
• [10] I. Pinelis, Optimal tail comparison based on comparison of moments, in: High Dimensional Probability (Oberwolfach, 1996), Progr. Probab. 43, Birkhauser, 1998, 297-314.
• [11] G. Pisier, Probabilistic methods in the geometry of Banach spaces, in: Probability and Analysis (Varenna, 1985), Lecture Notes in Math. 1206, Springer, 1986, 167-241.
• [12] G. R. Shorack and J. A. Wellner, Empirical Processes with Applications to Statistics, Wiley, New York, 1986.
• [13] M. Talagrand, Isoperimetry, logarithmic Sobolev inequalities on the discrete cube, and Margulis' graph connectivity theorem, Geom. Funct. Anal. 3 (1993), 295-314.
• [14] —, The Generic Chaining. Upper and Lower Bounds of Stochastic Processes, Springer Monogr. Math., Springer, Berlin, 2005.