  In this paper, we give explicit rates in the central limit theorem and in the
almost sure invariance principle for general R d-valued cocycles that appear in
the study of the left random walk on linear groups. Our method of proof lies on
a suitable martingale approximation and on a careful estimation of some
coupling coefficients linked with the underlying Markov structure. Concerning
the martingale part, the available results in the literature are not accurate
enough to give almost optimal rates whether in the central limit theorem for
the Wasserstein distance, or in the strong approximation. A part of this paper
is devoted to circumvent this issue. We then exhibit near optimal rates both in
the central limit theorem in terms of Wasserstein distance and in the almost
sure invariance principle for R d-valued martingales with stationary increments
having moments of order p $\in$]2, 3] (the case of sequences of reversed
martingale differences is also considered). Note also that, as an application
of our results for general R d-valued cocycles, a special attention is paid to
the Iwasawa cocycle and the Cartan projection for reductive Lie groups.
