Rates of convergence in the strong invariance principle under projective criteria

We give rates of convergence in the strong invariance principle for stationary sequences satisfying some projective criteria. The conditions are expressed in terms of conditional expectations of partial sums of the initial sequence. Our results apply to a large variety of examples, including mixing processes of different kinds. We present some applications to symmetric random walks on the circle, to functions of dependent sequences, and to a reversible Markov chain.