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We give rates of convergence in the strong invariance principle forstationary sequences satisfying some projective criteria. The conditions areexpressed in terms of conditional expectations of partial sums of the initialsequence. Our results apply to a large variety of examples, including mixingprocesses of different kinds. We present some applications to symmetric randomwalks on the circle, to functions of dependent sequences, and to a reversibleMarkov chain.

Rates of convergence in the strong invariance principle under projective criteria