The limit of the spectral radius of block Toeplitz matrices with nonnegative entries

A bi-infinite sequence ..., t(-2), t(-1), t(0), t(1), t(2), ... of nonnegat ive p x p matrices defines a sequence of block Toeplitz matrices T-n = (t(i k)), n = 1, 2, ...,, where t(ik) = t(k-i), i, k = 1, ..., n. Under certain irreducibility assumptions, we show that the limit of the spectral radius o f T-n, as n tends to infinity, is given by inf {sigma(xi) : xi epsilon [0, infinity]}, where sigma(xi) is the spectral radius of Sigma(j epsilon Z)t(j )xi(j).