Estimating the dimension of a linear system

The problem considered is that of estimating the integer or integers that prescribe the dimension of a linear system. These could be the Kronecker indices. Though attention is concentrated on the order or McMillan degree, which specifies the dimension of a minimal state vector, the same results are available for other cases. A fairly complete theorem is proved relating to conditions under which strong or weak convergence will hold for an estimate of the McMillan degree when the estimation is based on minimisation of a criterion of the form log det([Omega]n) + nC(T)/T, where [Omega]n, is the estimate of the prediction error covariance matrix and the McMillan degree is assumed to be n. The conditions relate to the prescribed sequence C(T).