Asymptotic properties of powers of nonnegative matrices, with applications (Q1118006)

The main result of this paper gives first order approximations to \(f_ n(A)\) as \(n\to \infty\), for certain sequences \(f_ n\) of analytic functions, where A is a finite square reducible matrix in Frobenius form. In particular it holds when \(f_ n(A)=A^ n\), and hence can be used to study local behaviour of entries as well as blocks of \(A^ n\). An application to the behaviour of the vector \(pA^ n\) (``multiplicative processes'') generalizes a theorem of \textit{P. Mandl} [Casopis Pĕst. Mat. 84, 140-149 (1959; Zbl 0087.136)] who considered only non-degenerate and aperiodic diagonal blocks. The approach is via spectral properties of finite dimensional linear operators.