Erdös–Rényi laws for dynamical systems

We establish Erdös–Rényi limit laws for Lipschitz observations on a class of non‐uniformly expanding dynamical systems, including logistic‐like maps. These limit laws give the maximal average of a time series over a time window of logarithmic length. We also give results on maximal averages of a time series arising from Hölder observations on intermittent‐type maps over a time window of polynomial length. We consider the rate of convergence in the limit law for subshifts of finite type and establish a one‐sided rate bound for Gibbs–Markov maps.

[1]  R. R. Bahadur,et al.  On Deviations of the Sample Mean , 1960 .

[2]  A. Rényi,et al.  On a new law of large numbers , 1970 .

[3]  E. Stein.  Singular Integrals and Di?erentiability Properties of Functions , 1971 .

[4]  J. Cooper.  SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONS , 1973 .

[5]  Stephen A. Book.  An extension of the Erdős-Rényi new law of large numbers , 1975 .

[6]  Pál Révész,et al.  How Big are the Increments of a Wiener Process , 1979 .

[7]  R. Ellis,et al.  Entropy, large deviations, and statistical mechanics , 1985 .

[8]  Luc Devroye,et al.  Exact Convergence Rate in the Limit Theorems of Erdos-Renyi and Shepp , 1986 .

[9]  D. Mason,et al.  An extended version of the Erdos-Renyi strong law of large numbers , 1989 .

[10]  C. Liverani,et al.  A probabilistic approach to intermittency , 1999, Ergodic Theory and Dynamical Systems.

[11]  Lai-Sang Young,et al.  Markov Extensions and Decay of Correlations for Certain Hénon Maps , 2000, Astérisque.

[12]  Pierre Collet,et al.  Statistics of closest return for some non-uniformly hyperbolic systems , 1999, Ergodic Theory and Dynamical Systems.

[13]  Jon Aaronson,et al.  LOCAL LIMIT THEOREMS FOR PARTIAL SUMS OF STATIONARY SEQUENCES GENERATED BY GIBBS–MARKOV MAPS , 2001 .

[14]  P. Collet,et al.  Almost-sure central limit theorems and the Erdös–Rényi law for expanding maps of the interval , 2002, Ergodic Theory and Dynamical Systems.

[15]  Zakhar Kabluchko,et al.  An Erdös-Rényi law for mixing processes , 2007 .

[16]  Sebastien Gouezel.  A Borel-Cantelli lemma for intermittent interval maps , 2007 .

[17]  Luc Rey-Bellet,et al.  Large deviations in non-uniformly hyperbolic dynamical systems , 2008, Ergodic Theory and Dynamical Systems.

[18]  Matthew Nicol,et al.  Large deviations for nonuniformly hyperbolic systems , 2008 .

[19]  Ian Melbourne,et al.  Large and moderate deviations for slowly mixing dynamical systems , 2008 .

[20]  Richard Sharp,et al.  Large deviations for intermittent maps , 2009 .