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

Abstract:

In this paper, we give explicit rates in the central limit theorem and in the almost sure invariance principle for general ${\mathbb R}^d$-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 either 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 the Wasserstein distance and in the almost sure invariance principle for ${\mathbb R}^d$-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 ${\mathbb R}^d$-valued cocycles, a special attention is paid to the Iwasawa cocycle and the Cartan projection for reductive Lie groups (like for instance ${\mathrm {GL}}_d(\mathbb {R})$).