  We propose some backward-forward martingale decompositions for functions of
reversible Markov chains. These decompositions are used to prove the functional
CLT for reversible Markov chains with asymptotically linear variance of partial
sums. We also provide a proof of the equivalence between asymptotic linearity
of the variance and convergence of the integral of $1/(1-t)$ with respect to
the associated spectral measure $\rho$. We also study the asymptotic behavior
of linear processes having as innovations mean zero square integrable functions
of stationary reversible Markov chains. We apply this study to several cases of
reversible stationary Markov chains that arise in regression estimation.
