The stochastic equation P(t+1)=A(t)P(t)+B(t) with non-stationary coefficients

In this paper we consider the stochastic sequence {Pt}E N defined recursively by the linear relation Pt+1 = At Pt + Bt in a random environment which is described by the non-stationary process V = {(At, Bt) t E N.. We formulate sufficient conditions on v which ensure that the finite-dimensional distributions of {Pt} E N converge weakly to the finite-dimensional distribution of a unique stationary process. If the driving sequence v has a nice tail behaviour, then we can establish a global convergence result. This extends results of Brandt (1986) and Borovkov (1998) from the stationary to the non-stationary ease.