This paper is concerned with the asymptotics for the sequence

${Y}_{t+1}={A}_{t}{Y}_{t}+{B}_{t}$ defined by the non-stationary random environment

${({A}_{t},{B}_{t})}_{t}$. Specifically, convergence in distribution is proven for the shifted process

${\left({Y}_{T+t}\right)}_{t}$ as

$T\to \infty $ under the condition that

${({A}_{t},{B}_{t})}_{t}$ is stationary under an auxiliary probability which coincides with the original probability on the tail field of

${({A}_{t},{B}_{t})}_{t}$. Moreover, convergence of finite-dimensional marginal distributions is shown to hold true also under the weaker assumption that the process

${({A}_{t},{B}_{t})}_{t}$ can be approximated by certain stationary and ergodic processes.