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

Rates of convergence in invariance principles for random walks on linear groups via martingale methods