  We extend the spectral approach of S. Gou\"ezel for the vector-valued almost
sure invariance principle (ASIP) to certain classes of non-stationary sequences
with a weaker control over the behavior of the covariance matrices, assuming
only linear growth. Then we apply this extension to obtain a quenched
vector-valued ASIP for random perturbations of a fixed Anosov diffeomorphism as
well as random perturbations of a billiard map associated to the periodic
Lorentz gas. We also consider certain classes of random piecewise expanding
maps.
