We prove the almost sure invariance principle for stationary ℝd-valued random processes (with very precise dimension-independent error terms), solely under a strong assumption concerning 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.
"Almost sure invariance principle for dynamical systems by spectral methods." Ann. Probab. 38 (4) 1639 - 1671, July 2010. https://doi.org/10.1214/10-AOP525