  We investigate a wide class of two-dimensional hyperbolic systems with
singularities, and prove the almost sure invariance principle (ASIP) for the
random process generated by sequences of dynamically H\"older observables. The
observables could be unbounded, and the process may be non-stationary and need
not have linearly growing variances. Our results apply to Anosov
diffeomorphisms, Sinai dispersing billiards and their perturbations. The random
processes under consideration are related to the fluctuation of Lyapunov
exponents, the shrinking target problem, etc.
