Consistent estimation of system order

We consider a parameterized family {S??,????A}, A??R??, of systems or sources having stochastic outputs {xn} that are partially described by a statistic (e.g., correlation function) ????(??). If we represent ?? = ??1, ??2,...,??n,..., then by the system order M?? we mean the index n of the last non-zero term in the expansion of ??. Our objective is to generate a sequence {Mn(x1,..., xn)} of estimates of the true M??0 that converge to it at least in probability. We provide conditions insuring the existence of such a statistically consistent sequence of estimators. We then apply our method to estimate the order of a scalar moving averages process, and the order of a scalar autoregressive process. The results we present are primarily of a theoretical nature