In this paper we survey some recent results on the central limit theorem and its weak invariance principle for stationary sequences. We also describe several maximal inequalities that are the main tool for obtaining the invariance principles, and also they have interest in themselves. The classes of dependent random variables considered will be martingale-like sequences, mixing sequences, linear processes, additive functionals of ergodic Markov chains.
"Recent advances in invariance principles for stationary sequences." Probab. Surveys 3 1 - 36, 2006. https://doi.org/10.1214/154957806100000202