Rates of convergence in the strong invariance principle for non adapted sequences. Application to ergodic automorphisms of the torus

TL;DR

This paper establishes convergence rates in the strong invariance principle for non-adapted sequences satisfying projective criteria, with applications to ergodic automorphisms of the torus, including non-hyperbolic cases.

Contribution

It provides explicit convergence rates for the strong invariance principle applied to iterates of ergodic automorphisms of the torus, extending to non-hyperbolic cases.

Findings

Applicable to a large class of unbounded functions on the torus

Provides explicit rates of convergence in the invariance principle

Extends results to non-hyperbolic ergodic automorphisms

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, even in the non hyperbolic case. In this context, we give a large class of unbounded functions f from the d-dimensional torus to R, for which the partial sum foT+ foT^2 + ... + foT^n satisfies a strong invariance principle with an explicit rate of convergence.