The concentration of measure phenomenon

Concentration functions and inequalities Isoperimetric and functional examples Concentration and geometry Concentration in product spaces Entropy and concentration Transportation cost inequalities Sharp bounds of Gaussian and empirical processes Selected applications References Index.

[1]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[2]  E. Schmidt Die Brunn-Minkowskische Ungleichung und ihr Spiegelbild sowie die isoperimetrische Eigenschaft der Kugel in der euklidischen und nichteuklidischen Geometrie. I , 1948 .

[3]  W. Hoeffding Probability Inequalities for sums of Bounded Random Variables , 1963 .

[4]  L. H. Harper Optimal numberings and isoperimetric problems on graphs , 1966 .

[5]  V. V. Jurinskii Exponential Bounds for Large Deviations , 1974 .

[6]  V. V. Yurinskii Exponential inequalities for sums of random vectors , 1976 .

[7]  I. Ibragimov,et al.  Norms of Gaussian sample functions , 1976 .

[8]  Da-Lun Wang,et al.  Extremal Configurations on a Discrete Torus and a Generalization of the Generalized Macaulay Theorem , 1977 .

[9]  V. Sudakov,et al.  Extremal properties of half-spaces for spherically invariant measures , 1978 .

[10]  J. Kuelbs Probability on Banach spaces , 1978 .

[11]  B. Simon Trace ideals and their applications , 1979 .

[12]  Zoltán Füredi,et al.  A short proof for a theorem of Harper about Hamming-spheres , 1981, Discret. Math..

[13]  O. Rothaus Diffusion on compact Riemannian manifolds and logarithmic Sobolev inequalities , 1981 .

[14]  G. Kallianpur Stochastic differential equations and diffusion processes , 1981 .

[15]  M. Gromov,et al.  A topological application of the isoperimetric inequality , 1983 .

[16]  Gilles Pisier,et al.  On the dimension of the ⁿ_{}-subspaces of Banach spaces, for 1≤<2 , 1983 .

[17]  Andrzej Korzeniowski,et al.  An example in the theory of hypercontractive semigroups , 1985 .

[18]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[19]  S. Rachev The Monge–Kantorovich Mass Transference Problem and Its Stochastic Applications , 1985 .

[20]  M. Pratelli,et al.  Probability and Analysis , 1986 .

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

[22]  V. Milman Diameter of a minimal invariant subset of equivariant lipschitz actions on compact subsets of ℝ k , 1987 .

[23]  M. Gromov,et al.  Generalization of the spherical isoperimetric inequality to uniformly convex Banach spaces , 1987 .

[24]  M. Talagrand An isoperimetric theorem on the cube and the Kintchine-Kahane inequalities , 1988 .

[25]  Terry Lyons,et al.  A crossing estimate for the canonical process on a Dirichlet space and tightness result , 1988 .

[26]  M. Talagrand Isoperimetry and Integrability of the Sum of Independent Banach-Space Valued Random Variables , 1989 .

[27]  Colin McDiarmid,et al.  Surveys in Combinatorics, 1989: On the method of bounded differences , 1989 .

[28]  竹田 雅好 On a martingale method for symmetric diffusion processes and its applications , 1989 .

[29]  G. Schechtman A remark concerning the dependence on ɛ in dvoretzky's theorem , 1989 .

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

[31]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[32]  Gideon Schechtman,et al.  On the volume of the intersection of two $L\sp n\sb p$ balls , 1990 .

[33]  M. Talagrand A new isoperimetric inequality and the concentration of measure phenomenon , 1991 .

[34]  Michel Talagrand A new isoperimetric inequality for product measure and the tails of sums of independent random variables , 1991 .

[35]  Stanisław Kwapień,et al.  Hypercontraction Methods in Moment Inequalities for Series of Independent Random Variables in Normed Spaces , 1991 .

[36]  G. Schechtman,et al.  Remarks on Talagrand’s deviation inequality for Rademacher functions , 1990, math/9201208.

[37]  Jim Freeman Probability Metrics and the Stability of Stochastic Models , 1991 .

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

[39]  Vitali Milman,et al.  Dvoretzky's theorem — Thirty years later , 1992 .

[40]  M. Ledoux,et al.  A heat semigroup approach to concentration on the sphere and on a compact Riemannian manifold , 1992 .

[41]  S. Kwapień,et al.  Random Series and Stochastic Integrals: Single and Multiple , 1992 .

[42]  Miklós Simonovits,et al.  Random Walks in a Convex Body and an Improved Volume Algorithm , 1993, Random Struct. Algorithms.

[43]  H. Kesten On the Speed of Convergence in First-Passage Percolation , 1993 .

[44]  M. Talagrand THE SUPREMUM OF SOME CANONICAL PROCESSES , 1994 .

[45]  M. Talagrand Concentration of measure and isoperimetric inequalities in product spaces , 1994, math/9406212.

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

[47]  M. Schmuckenschläger A Concentration of Measure Phenomenon on Uniformly Convex Bodies , 1995 .

[48]  D. Nualart The Malliavin Calculus and Related Topics , 1995 .

[49]  Michel Ledoux Remarks on logarithmic Sobolev constants, exponential integrability and bounds on the diameter , 1995 .

[50]  Miklós Simonovits,et al.  Isoperimetric problems for convex bodies and a localization lemma , 1995, Discret. Comput. Geom..

[51]  G. Schechtman,et al.  A CONCENTRATION INEQUALITY FOR HARMONIC MEASURES ON THE SPHERE , 1995 .

[52]  D. Stroock,et al.  Probability Theory: An Analytic View. , 1995 .

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

[54]  M. Talagrand New concentration inequalities in product spaces , 1996 .

[55]  K. Marton A measure concentration inequality for contracting markov chains , 1996 .

[56]  M. Talagrand Transportation cost for Gaussian and other product measures , 1996 .

[57]  K. Marton Bounding $\bar{d}$-distance by informational divergence: a method to prove measure concentration , 1996 .

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

[59]  M. Ledoux,et al.  Isoperimetry and Gaussian analysis , 1996 .

[60]  M. Talagrand A new look at independence , 1996 .

[61]  Krzysztof Oleszkiewicz,et al.  Comparison of Moments of Sums of Independent Random Variables and Differential Inequalities , 1996 .

[62]  M. Ledoux On Talagrand's deviation inequalities for product measures , 1997 .

[63]  L. Saloff-Coste,et al.  Lectures on finite Markov chains , 1997 .

[64]  M. Schmuckenschläger Martingales, Poincaré Type Inequalities, and Deviation Inequalities , 1998 .

[65]  M. Schmuckenschläger Curvature of Nonlocal Markov Generators , 1998 .

[66]  M. Ledoux The geometry of Markov diffusion generators , 1998 .

[67]  O. Rothaus Logarithmic Sobolev inequalities and the growth of $L^p$ norms , 1998 .

[68]  Michel Talagrand,et al.  The Sherrington–Kirkpatrick model: a challenge for mathematicians , 1998 .

[69]  K. Marton Measure concentration for a class of random processes , 1998 .

[70]  C. Houdré,et al.  Interpolation, correlation identities, and inequalities for infinitely divisible variables , 1998 .

[71]  Elton P. Hsu Analysis on path and loop spaces , 1999 .

[72]  N. Yoshida The log-Sobolev inequality for weakly coupled lattice fields , 1999 .

[73]  M. Ledoux Concentration of measure and logarithmic Sobolev inequalities , 1999 .

[74]  M. Gromov Metric Structures for Riemannian and Non-Riemannian Spaces , 1999 .

[75]  V. Pestov Amenable representations and dynamics of the unit sphere in an infinite-dimensional Hilbert space , 1999, math/9903085.

[76]  Olivier Guédon Kahane-Khinchine type inequalities for negative exponent , 1999 .

[77]  A. Soshnikov Universality at the Edge of the Spectrum¶in Wigner Random Matrices , 1999, math-ph/9907013.

[78]  G. Royer,et al.  Une initiation aux inégalités de Sobolev logarithmiques , 1999 .

[79]  E. Rio Inégalités de Hoeffding pour les fonctions lipschitziennes de suites dépendantes , 2000 .

[80]  LpWilliam B. Johnson,et al.  Finite dimensional subspaces of , 2000 .

[81]  Liming Wu,et al.  A new modified logarithmic Sobolev inequality for Poisson point processes and several applications , 2000 .

[82]  C. Villani,et al.  Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality , 2000 .

[83]  P. Massart Some applications of concentration inequalities to statistics , 2000 .

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

[85]  A. Guionnet,et al.  CONCENTRATION OF THE SPECTRAL MEASURE FOR LARGE MATRICES , 2000 .

[86]  Vladimir Pestov,et al.  Ramsey-Milman phenomenon, Urysohn metric spaces, and extremely amenable groups , 2000, math/0004010.

[87]  G. Schechtman An editorial comment on the preceding paper , 2000 .

[88]  V. Milman,et al.  Chapter 17 - Euclidean Structure in Finite Dimensional Normed Spaces , 2001 .

[89]  Michel Ledoux,et al.  Logarithmic Sobolev Inequalities for Unbounded Spin Systems Revisited , 2001 .

[90]  Prasad Tetali,et al.  Concentration of Measure for Products of Markov Kernels and Graph Products via Functional Inequalities , 2001, Combinatorics, Probability and Computing.

[91]  J. Lindenstrauss,et al.  Handbook of geometry of Banach spaces , 2001 .

[92]  E. Rio,et al.  Inégalités de concentration pour les processus empiriques de classes de parties , 2001 .

[93]  Colin McDiarmid Concentration For Independent Permutations , 2002, Comb. Probab. Comput..

[94]  E. Rio Une inégalité de Bennett pour les maxima de processus empiriques , 2002 .

[95]  V. Paulauskas Some Comments on Deviation Inequalities for Infinitely Divisible Random Vectors , 2002 .

[96]  C. Houdré Remarks on deviation inequalities for functions of infinitely divisible random vectors , 2002 .

[97]  Alice Guionnet,et al.  Lectures on Logarithmic Sobolev Inequalities , 2003 .