Exponential inequalities for dynamical measures of expanding maps of the interval

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 inequality. We apply these results to the estimation of the rate of convergence of the empirical measure in various metrics and also to the efficiency of the shadowing by sets of positive measure.

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

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

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

[4]  A. Barbour,et al.  Iterates of expanding maps , 2000 .

[5]  C. Liverani.  Decay of correlations , 1995 .

[6]  J. Buzzi.  Specification on the interval , 1997 .

[7]  Franz Hofbauer,et al.  Ergodic properties of invariant measures for piecewise monotonic transformations , 1982 .

[8]  P. Massart,et al.  About the constants in Talagrand's concentration inequalities for empirical processes , 2000 .

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

[10]  Estimation non paramétrique pour des processus dynamiques dilatants , 2000 .

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

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

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

[14]  F. Hofbauer.  Piecewise invertible dynamical systems , 1986 .

[15]  V. Maume-Deschamps.  Correlation decay for Markov maps on a countable state space , 2001, Ergodic Theory and Dynamical Systems.

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

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

[18]  J. Yorke,et al.  On the existence of invariant measures for piecewise monotonic transformations , 1973 .

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