Rates of Convergence in the Strong Invariance Principle for Non-adapted Sequences Application to Ergodic Automorphisms of the Torus

Abstract

In this paper, we give rates of convergence in the strong invariance principle for non-adapted sequences satisfying projective criteria. The results apply to the iterates of ergodic automorphisms T of the d-dimensional torus \(\mathbb{T}^d\), even in the non hyperbolic case. In this context, we give a large class of unbounded function f from \(\mathbb{T}^d\) to \(\mathbb{R}\), for which the partial sum \(f\;\circ\;T\;+\;f\;\circ\;T^{2}\;+\;.\;.\;.\;+\;f\;\circ\;T^{n}\) satisfies a strong invariance principle with an explicit rate of convergence.