We consider an i.i.d. random environment with a strong form of transience on the two dimensional integer lattice. Namely, the walk always moves forward in the y-direction. We prove an invariance principle for the quenched expected position of the random walk indexed by its level crossing times. We begin with a variation of the Martingale Central Limit Theorem. The main part of the paper checks the conditions of the theorem for our problem.
"Fluctuations of the quenched mean of a planar random walk in an i.i.d. random environment with forbidden direction." Electron. J. Probab. 14 1268 - 1289, 2009. https://doi.org/10.1214/EJP.v14-655