  In this paper we survey the almost sure central limit theorem and its
functional form (quenched) for stationary and ergodic processes. For additive
functionals of a stationary and ergodic Markov chain these theorems are known
under the terminology of central limit theorem and its functional form, started
at a point. All these results have in common that they are obtained via a
martingale approximation in the almost sure sense. We point out several
applications of these results to classes of mixing sequences, shift processes,
reversible Markov chains, Metropolis Hastings algorithms.
