Let $X$ be a Harris recurrent Markov process (in discrete or continuous time). We give a functional law of the iterated logarithm for the additive functionals of $X$ which are (close to) square integrable martingales with respect to the invariant measure of $X$. The proof is based on the Skorokhod embedding technique and the construction of an atom for a Harris chain. In contrast with the positive recurrent case, "the suitable normalizations" are random in the null recurrent case. Moreover it is shown from two examples how to use the law of the iterated logarithm to get the rate of almost sure convergence of an estimator.
Abderrahmen Touati. "Loi Fonctionnelle du Logarithme Itere Pour les Processus de Markov Recurrents." Ann. Probab. 18 (1) 140 - 159, January, 1990. https://doi.org/10.1214/aop/1176990942