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

Law of large numbers for random walk with unbounded jumps and BDP with bounded jumps in random environment