A lowerbound on the dimension of positive realizations

A basic phenomenon in positive system theory is that the dimension N of an arbitrary positive realization of a given transfer function H(z) may be strictly larger than the dimension n of its minimal realizations. The aim of this brief is to provide a nontrivial lowerbound on the value of N under the assumption that there exists a time instant k₀ at which the (always nonnegative) impulse response of H(z) is 0 but the impulse response becomes strictly positive for all k>k₀. Transfer functions with this property may be regarded as extremal cases in positive system theory.