  We consider a ballistic random walk in an i.i.d. random environment that does
not allow retreating in a certain fixed direction. Homogenization and
regeneration techniques combine to prove a law of large numbers and an averaged
invariance principle. The assumptions are non-nestling and $1+\e$ (resp.\
$2+\e$) moments for the step of the walk uniformly in the environment, for the
law of large numbers (resp. invariance principles). We also investigate
invariance principles under fixed environments, and invariance principles for
the environment-dependent mean of the walk.
