This paper is concerned with the asymptotics for the sequence
defined by the non-stationary random environment
. Specifically, convergence in distribution is proven for the shifted process
under the condition that
is stationary under an auxiliary probability which coincides with the original probability on the tail field of
. Moreover, convergence of finite-dimensional marginal distributions is shown to hold true also under the weaker assumption that the process
can be approximated by certain stationary and ergodic processes.