Rates of convergence in the strong invariance principle for non adapted   sequences. Application to ergodic automorphisms of the torus
  In this paper, we give rates of convergence in the strong invariance principle for non-adapted sequences satisfying projective criteria. The results apply to the iterates of ergodic automorphisms T of the d-dimensional torus, even in the non hyperbolic case. In this context, we give a large class of unbounded functions f from the d-dimensional torus to R, for which the partial sum foT+ foT^2 + ... + foT^n satisfies a strong invariance principle with an explicit rate of convergence. 
J. Dedecker, F. Merlev\`ede and F. P\`ene
math.PR