A new look at independence

The concentration of measure phenomenon in product spaces is a farreaching abstract generalization of the classical exponential inequalities for sums of independent random variables. We attempt to explain in the simplest possible terms the basic concepts underlying this phenomenon, the basic method to prove concentration inequalities and the meaning of several of the most useful inequalities.

[1]  A. Ehrhard.  Symétrisation dans l'espace de Gauss. , 1983 .

[2]  Joel H. Spencer,et al.  Sharp concentration of the chromatic number on random graphsGn, p , 1987, Comb..

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

[4]  N. Fisher,et al.  Probability Inequalities for Sums of Bounded Random Variables , 1994 .

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

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

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

[8]  V. Milman,et al.  Unconditional and symmetric sets inn-dimensional normed spaces , 1980 .

[9]  M. Talagrand.  Isoperimetry, logarithmic sobolev inequalities on the discrete cube, and margulis' graph connectivity theorem , 1993 .

[10]  J J Hopfield,et al.  Neural networks and physical systems with emergent collective computational abilities. , 1982, Proceedings of the National Academy of Sciences of the United States of America.

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

[12]  Béla Bollobás,et al.  The Height of a Random Partial Order: Concentration of Measure , 1992 .

[13]  M. Talagrand.  A conjecture on convolution operators, and a non-Dunford-Pettis operator onL1 , 1989 .

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

[15]  Katalin Marton,et al.  A simple proof of the blowing-up lemma , 1986, IEEE Trans. Inf. Theory.

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

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

[18]  Leonid Pastur,et al.  Absence of self-averaging of the order parameter in the Sherrington-Kirkpatrick model , 1991 .

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

[20]  W. Thurston.  On Proof and Progress in Mathematics , 1994, math/9404236.

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