The stochastic equation Yt+1 = AtYt + Bt with non-stationary coefficients

Abstract

In this paper, we consider the stochastic sequence {Y_t}_t∊ℕ defined recursively by the linear relation Y_t+1 = A_tY_t + B_t in a random environment which is described by the non-stationary process {(A_t, B_t)}_t∊ℕ. We formulate sufficient conditions on the environment which ensure that the finite-dimensional distributions of {Y_t}_t∊ℕ converge weakly to the finite-dimensional distributions of a unique stationary process. If the driving sequence {(A_t, B_t)}_t∊ℕ becomes 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.