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We establish almost sure invariance principles, a strong form ofapproximation by Brownian motion, for non-stationary time-series arising asobservations on dynamical systems. Our examples include observations onsequential expanding maps, perturbed dynamical systems, non-stationarysequences of functions on hyperbolic systems as well as applications to theshrinking target problem in expanding systems.

Almost sure invariance principle for sequential and non-stationary dynamical systems