  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 a functional CLT 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.
