Given an ergodic dynamical system (X, T, µ), and U ⊂ X measurable with µ(U) > 0, let µ(U)τ U (x) denote the normalized hitting time of x ∈ X to U. We prove that given a sequence (U n ) with µ(U n ) → 0, the distribution function of the normalized hitting times to U n converges weakly to some sub-probability distribution F if and only if the distribution function of the normalized return time converges weakly to some distribution functionF , and that in the converging case, F (t) = t 0 2000 Mathematics Subject Classification 37A05, 37A50, 60F05, 28D05.
We establish almost sure invariance principles, a strong form of approximation by Brownian motion, for non-stationary time-series arising as observations on dynamical systems. Our examples include observations on sequential expanding maps, perturbed dynamical systems, non-stationary sequences of functions on hyperbolic systems as well as applications to the shrinking target problem in expanding systems.
Previously it has been shown that some classes of mixing dynamical systems have limiting return times distributions that are almost everywhere Poissonian. Here we study the behaviour of return times at periodic points and show that the limiting distribution is a compound Poissonian distribution. We also derive error terms for the convergence to the limiting distribution. We also prove a very general theorem that can be used to establish compound Poisson distributions in many other settings.
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