The stochastic equationYt+1=AtYt+Btwith non-stationary coefficients

In this paper, we consider the stochastic sequence {Yt}t∊ℕdefined recursively by the linear relationYt+1=AtYt+Btin a random environment which is described by the non-stationary process {(At,Bt)}t∊ℕ. We formulate sufficient conditions on the environment which ensure that the finite-dimensional distributions of {Yt}t∊ℕconverge weakly to the finite-dimensional distributions of a unique stationary process. If the driving sequence {(At,Bt)}t∊ℕbecomes stationary in the long run, then we can establish a global convergence result. This extends results of Brandt (1986) and Borovkov (1998) from the stationary to the non-stationary case.