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.
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