Almost-sure central limit theorems and the Erdös-Rényi law for expanding maps of the interval

For a large class of expanding maps of the interval, we prove that partial sums of Lipschitz observables satisfy an almost-sure central limit theorem (ASCLT). In fact, we provide a rate of convergence in the Kantorovich distance. Maxima of partial sums are also shown to obey an ASCLT. The key tool is an exponential inequality recently obtained. Then we establish (optimal) almost-sure convergence rates for the supremum of moving averages of Lipschitz observables (Erdös-Ŕnyi-type law). This is done by refining the usual large-deviations estimates available for expanding maps of the interval. We end up with an application to entropy estimation ASCLTs that refine the Shannon-McMillan-Breiman and Ornstein-Weiss theorems.