  Consider a random walk in a time-dependent random environment on the lattice
Zd. Recently, Rassoul-Agha, Seppalainen and Yilmaz [RSY11] proved a general
large deviation principle under mild ergodicity assumptions on the random
environment for such a random walk, establishing first level 2 and 3 large
deviation principles. Here we present two alternative short proofs of the level
1 large deviations under mild ergodicity assumptions on the environment: one
for the continuous time case and another one for the discrete time case. Both
proofs provide the existence, continuity and convexity of the rate function.
Our methods are based on the use of the sub-additive ergodic theorem as
presented by Varadhan in 2003.
