Weak convergence of a random walk in a random environment

Let _i (x),i=1,...,d,xZ ^d , satisfy _i (x)>0, and ₁(x)+...+ _d (x)=1. Define a Markov chain onZ ^d by specifying that a particle atx takes a jump of +1 in thei ^th direction with probability 1/2 _i (x) and a jump of –1 in thei ^th direction with probability 1/2 _i (x). If the _i (x) are chosen from a stationary, ergodic distribution, then for almost all the corresponding chain converges weakly to a Brownian motion.