Almost sure invariance principle for dynamical systems by spectral methods

TL;DR

This paper establishes an almost sure invariance principle for stationary vector-valued processes using spectral methods, providing precise error bounds under verifiable spectral assumptions, applicable to various dynamical systems and Markov chains.

Contribution

It introduces a spectral approach to prove the invariance principle with dimension-independent error terms under a checkable characteristic function condition.

Findings

Validates the invariance principle for broad classes of systems

Provides dimension-independent error estimates

Uses spectral perturbation techniques for verification

Abstract

We prove the almost sure invariance principle for stationary R^d--valued processes (with dimension-independent very precise error terms), solely under a strong assumption on the characteristic functions of these processes. This assumption is easy to check for large classes of dynamical systems or Markov chains, using strong or weak spectral perturbation arguments.