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We investigate a wide class of two-dimensional hyperbolic systems withsingularities, and prove the almost sure invariance principle (ASIP) for therandom process generated by sequences of dynamically H\"older observables. Theobservables could be unbounded, and the process may be non-stationary and neednot have linearly growing variances. Our results apply to Anosovdiffeomorphisms, Sinai dispersing billiards and their perturbations. The randomprocesses under consideration are related to the fluctuation of Lyapunovexponents, the shrinking target problem, etc.

Non-stationary Almost Sure Invariance Principle for Hyperbolic Systems with Singularities