Concentration inequalities for random fields via coupling

We present a new and simple approach to concentration inequalities in the context of dependent random processes and random fields. Our method is based on coupling and does not use information inequalities. In case one has a uniform control on the coupling, one obtains exponential concentration inequalities. If such a uniform control is no more possible, then one obtains polynomial or stretched-exponential concentration inequalities. Our abstract results apply to Gibbs random fields, both at high and low temperatures and in particular to the low-temperature Ising model which is a concrete example of non-uniformity of the coupling.

[1]  P. Heywood.  Trigonometric Series , 1968, Nature.

[2]  Adriano M. Garsia,et al.  Martingale inequalities: seminar notes on recent progress , 1973 .

[3]  H. Poincaré,et al.  Percolation ? , 1982 .

[4]  Hans-Otto Georgii,et al.  Gibbs Measures and Phase Transitions , 1988 .

[5]  R. M. Dudley,et al.  Real Analysis and Probability , 1989 .

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

[7]  L. Devroye.  Exponential Inequalities in Nonparametric Estimation , 1991 .

[8]  T. Lindvall.  Lectures on the Coupling Method , 1992 .

[9]  Van den Berg,et al.  Disagreement percolation in the study of Markov fields , 1994 .

[10]  D. Ioffe.  Exact large deviation bounds up toTc for the Ising model in two dimensions , 1995 .

[11]  Z. Su.  Central limit theorems for random processes with sample paths in exponential Orlicz spaces , 1997 .

[12]  J. Yukich.  Probability theory of classical Euclidean optimization problems , 1998 .

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

[14]  S. Bobkov,et al.  Exponential Integrability and Transportation Cost Related to Logarithmic Sobolev Inequalities , 1999 .

[15]  The random geometry of equilibrium phases , 1999, math/9905031.

[16]  Paul-Marie Samson,et al.  Concentration of measure inequalities for Markov chains and $\Phi$-mixing processes , 2000 .

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

[18]  Percolation, Path Large Deviations and Weakly Gibbs States , 2000 .

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

[20]  Luc Devroye,et al.  Combinatorial methods in density estimation , 2001, Springer series in statistics.

[21]  P. Collet,et al.  Exponential inequalities for dynamical measures of expanding maps of the interval , 2002 .

[22]  Malempati M. Rao,et al.  Applications Of Orlicz Spaces , 2002 .

[23]  C. Külske.  Concentration Inequalities for Functions of Gibbs Fields with Application to Diffraction and Random Gibbs Measures , 2003 .

[24]  K. Marton.  Measure concentration and strong mixing , 2003 .

[25]  Devroye inequality for a class of non-uniformly hyperbolic dynamical systems , 2004, math/0412166.

[26]  P. Collet,et al.  Statistical consequences of the Devroye inequality for processes. Applications to a class of non-uniformly hyperbolic dynamical systems , 2004, math/0412167.