Abstract: The author extends the cross-estimate of Lyons-Zheng Markov processes from the symmetric case to the general stationary situation. The main tool is the forward-backward martingale decomposition. As a corollary the author obtains a tightness result for laws of Markov processes. Finally the author gives an example to show how to obtain the crucial cross-estimate.

Key words: Forward-backward martingale decomposition, Cross-estimate, Quasi-symmetric, Tightness