Optimal Concentration Inequalities for Dynamical Systems

被引：30

作者：

Chazottes, Jean-Rene ^{[1
]}

Gouezel, Sebastien ^{[2
]}

机构：

[1] Ecole Polytech, CNRS, CPHT, UMR 7644, F-91128 Palaiseau, France

[2] Univ Rennes 1, IRMAR, CNRS, UMR 6625, F-35042 Rennes, France

来源：

COMMUNICATIONS IN MATHEMATICAL PHYSICS | 2012年 / 316卷 / 03期

关键词：

DEVROYE INEQUALITY; RANDOM-VARIABLES; MIXING RATES; MAPS; DECAY; BILLIARDS; INTERVAL; MOMENT;

D O I：

10.1007/s00220-012-1596-7

中图分类号：

O4 [物理学];

学科分类号：

0702 ;

摘要：

For dynamical systems modeled by a Young tower with exponential tails, we prove an exponential concentration inequality for all separately Lipschitz observables of n variables. When tails are polynomial, we prove polynomial concentration inequalities. Those inequalities are optimal. We give some applications of such inequalities to specific systems and specific observables.

引用

页码：843 / 889

页数：47

共 21 条

[1]

[Anonymous], LECT NOTES MATH

[2]

Benedicks M, 2000, ASTERISQUE, P13

[3] Moment inequalities for functions of independent random variables [J].

Boucheron, S ;

Bousquet, O ;

Lugosi, G ;

Massart, P .

ANNALS OF PROBABILITY, 2005, 33 (02) :514-560

[4] DISTRIBUTION FUNCTION INEQUALITIES FOR MARTINGALES [J].

BURKHOLDER, DL .

ANNALS OF PROBABILITY, 1973, 1 (01) :19-42

[5] A concentration inequality for interval maps with an indifferent fixed point [J].

Chazottes, J. -R. ;

Collet, P. ;

Redig, F. ;

Verbitskiy, E. .

ERGODIC THEORY AND DYNAMICAL SYSTEMS, 2009, 29 :1097-1117

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

Chazottes, JR ;

Collet, P ;

Schmitt, B .

NONLINEARITY, 2005, 18 (05) :2341-2364

[7] Devroye inequality for a class of non-uniformly hyperbolic dynamical systems [J].

Chazottes, JR ;

Collett, P ;

Schmitt, B .

NONLINEARITY, 2005, 18 (05) :2323-2340

[8] A family of chaotic billiards with variable mixing rates [J].

Chernov, N ;

Zhang, HK .

STOCHASTICS AND DYNAMICS, 2005, 5 (04) :535-553

[9] Billiards with polynomial mixing rates [J].

Chernov, N ;

Zhang, HK .

NONLINEARITY, 2005, 18 (04) :1527-1553

[10] Exponential inequalities for dynamical measures of expanding maps of the interval [J].

Collet, P ;

Martinez, S ;

Schmitt, B .

PROBABILITY THEORY AND RELATED FIELDS, 2002, 123 (03) :301-322

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