Functional law of iterated logarithm for additive functionals of reversible Markov processes

Using the forward-backward martingale decomposition and the martingale limit theorems, we establish the functional law of iterated logarithm for an additive functional (At) of a reversible Markov process, under theminimal condition that\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\sigma ^2 (A) = \mathop {\lim }\limits_{t \to \infty } EA_t^2 /t$$ \end{document} exists inR. We extend also the previous remarkable functional central limit theorem of Kipnis and Varadhan.