A monotonicity property for random walk in a partially random environment

We prove a law of large numbers for random walks in certain kinds of i.i.d. random environments in Zd that is an extension of a result of Bolthausen et al. (2003) [4]. We use this result, along with the lace expansion for self-interacting random walks, to prove a monotonicity result for the first coordinate of the speed of the random walk under some strong assumptions on the distribution of the environment.