Martingale approximations for continuous-time and discrete-time stationary Markov processes

We show that the method of Kipnis and Varadhan [Comm. Math. Phys. 104 (1986) 1–19] to construct a Martingale approximation to an additive functional of a stationary ergodic Markov process via the resolvent is universal in the sense that a martingale approximation exists if and only if the resolvent representation converges. A sufficient condition for the existence of a martingale approximation is also given. As examples we discuss moving average processes and processes with normal generator.