Asymptotic Behavior of a Nonhomogeneous Linear Recurrence System1

Consider the nonhomogeneous linear recurrence system xn+1 = A + Bn xn + gn where A and Bn n = 0 1 are square matrices and gn n = 0 1 are column vectors. In this paper, we describe, in terms of the initial condition, the asymptotic behavior of the solutions of this equation in the case when A has a simple dominant eigenvalue λ0 ∞ n=0 Bn < ∞ and ∞ n=0 λ0 −n gn < ∞. The proof is based on the duality between the solutions of the above equation and the solutions of the associated adjoint equation. As a consequence, we obtain a similar result for higher order scalar equations.