  We study random walk with unbounded jumps in random environment. The
environment is stationary and ergodic, uniformly elliptic and decays
polynomially with speed $Dj^{-(3+\varepsilon_0)}$ for some small
$\varepsilon_0>0$ and proper $D>0.$ We prove a law of large number with
positive velocity under the condition that the annealed mean of the hitting
time of the positive half lattice is finite. Secondly, we consider birth and
death process with bounded jumps in stationary and ergodic environment. Under
the uniformly elliptic condition, we prove a law of large number and give the
explicit formula of its velocity.
