Weighted limits for Poincaré difference equations

The Poincaré difference equation xn+1=Anxn,n∈N, is considered, where AnAn, n∈Nn∈N, are complex square matrices such that the limit A=limn→∞AnA=limn→∞An exists. It is shown that under appropriate spectral conditions certain weighted limits of the nonvanishing solutions exist. In the case when the entries of the coefficients AnAn, n∈Nn∈N, and the initial vector  x0 are real our result implies the convergence of the normalized sequence xn‖xn‖, n∈Nn∈N, to a normalized eigenvector of the limiting matrix  AA.