Consistent estimation of system order

We consider a parameterized family , of systems or sources having stochastic outputs that are partially described by a statistic (e.g, correlation function) . If we represent , then by the system order M_{alpha} we mean the index of the last no nonzero term in the expansinn of α. Our objective is to generate a sequence of estimates of the that converge to it at least in probability. We provide conditions ensuring the existence of such a statistically consistent sequence of estimators, as wen as improved conditions yielding convergence in mean-square and with probability one. We establish existence by providing a method for constructing a family of consistent estimators of system order. We then apply our method to estimate the order of a scalar moving averages process and the order of a scalar autoregressive process. Our present results are primarily Of a theoretical nature, as we lack the efficiency and simulation studies desirable in support of a practical estimator of system order.