# Concentration inequalities for random fields via coupling

@article{Chazottes2005ConcentrationIF, title={Concentration inequalities for random fields via coupling}, author={J. R. Chazottes and P. Collet and Christof K{\"u}lske and Frank Redig}, journal={Probability Theory and Related Fields}, year={2005}, volume={137}, pages={201-225} }

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…

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## References

SHOWING 1-10 OF 33 REFERENCES

Concentration Inequalities for Functions of Gibbs Fields with Application to Diffraction and Random Gibbs Measures

- Mathematics
- 2003

We derive useful general concentration inequalities for functions of Gibbs fields in the uniqueness regime. We also consider expectations of random Gibbs measures that depend on an additional…

Concentration of measure inequalities for Markov chains and $\Phi$-mixing processes

- Mathematics
- 2000

We prove concentration inequalities for some classes of Markov chains and (F-mixing processes, with constants independent of the size of the sample, that extend the inequalities for product measures…

Percolation, Path Large Deviations and Weakly Gibbs States

- Mathematics
- 2000

Abstract: We present a unified approach to establishing the Gibbsian character of a wide class of non-Gibbsian states, arising in the Renormalisation Group theory. Inside the realm of the…

The random geometry of equilibrium phases

- Mathematics, Physics
- 1999

Publisher Summary This chapter discusses the random geometry of equilibrium phases. Percolation will come into play here on various levels. Its concepts like clusters, open paths, connectedness etc.…

Statistical consequences of the Devroye inequality for processes. Applications to a class of non-uniformly hyperbolic dynamical systems

- Mathematics
- 2005

In this paper, we apply the Devroye inequality to study various statistical estimators and fluctuations of observables for processes. Most of these observables are suggested by dynamical systems.…

Devroye inequality for a class of non-uniformly hyperbolic dynamical systems

- Mathematics
- 2005

In this paper we prove an inequality which we call the 'Devroye inequality' for a large class of non-uniformly hyperbolic dynamical systems (M, f). This class, introduced by Young, includes families…

Central limit theorems for random processes with sample paths in exponential Orlicz spaces

- Mathematics
- 1997

This paper is devoted to the study of central limit theorems and the domain of normal attraction for some random processes with sample paths in exponential Orlicz spaces under metric entropy…

Percolation ?

- 1982

572 NOTICES OF THE AMS VOLUME 53, NUMBER 5 Percolation is a simple probabilistic model which exhibits a phase transition (as we explain below). The simplest version takes place on Z2, which we view…

Exponential inequalities for dynamical measures of expanding maps of the interval

- Mathematics
- 2002

Abstract. We prove an exponential inequality for the absolutely continuous invariant measure of a piecewise expanding map of the interval. As an immediate corollary we obtain a concentration…

Gibbs Measures and Phase Transitions

- Computer Science
- 1988

This comprehensive monograph covers in depth a broad range of topics in the mathematical theory of phase transition in statistical mechanics and serves both as an introductory text and as a reference for the expert.