  We prove a law of large numbers for random walks in certain kinds of i.i.d.
random environments in Z^d that is an extension of a result of Bolthausen,
Sznitman and Zeitouni (2003). 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.
