We derive a new maximal inequality for stationary sequences under a martingale-type condition introduced by Maxwell and Woodroofe [Ann. Probab. 28 (2000) 713–724]. Then, we apply it to establish the Donsker invariance principle for this class of stationary sequences. A Markov chain example is given in order to show the optimality of the conditions imposed.
"A new maximal inequality and invariance principle for stationary sequences." Ann. Probab. 33 (2) 798 - 815, March 2005. https://doi.org/10.1214/009117904000001035