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.

中文翻译：

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通过鞅方法在线性群上随机游走的不变性原理的收敛率

在本文中，我们在中心极限定理和几乎肯定不变性原理中给出了在线性群的左随机游走研究中出现的一般 R d 值共环的显式速率。我们的证明方法依赖于合适的鞅近似以及对与基础马尔可夫结构相关联的一些耦合系数的仔细估计。关于鞅部分，无论是在 Wasserstein 距离的中心极限定理中还是在强近似中，文献中的可用结果都不够准确，无法给出几乎最佳的速率。本文的一部分专门用于规避这个问题。然后，我们在根据 Wasserstein 距离的中心极限定理和几乎确定的不变性原理中展示了接近最优的速率，对于 R d 值鞅，具有阶矩为 p $\in$]2, 3] 的平稳增量的 R d 值鞅（情况还考虑了反向鞅差异序列）。另请注意，作为我们对一般 R d 值共环的结果的应用，特别注意 Iwasawa 共环和还原李群的 Cartan 投影。