On the non-stationary covariance realization problem

Let z1 T denote the random vector of T distinct p-component output values of the non-stationary output sample z1 ?? of a linear time invariant stochastic system with state dimension d. When the parameterized covariance matrix of z1 T is denoted by ??T(??), for ?? ?? ?? ?? IR??, we say that ?? is identifiable (T, ??) if the map ??T(??): ?? ?? IR?? (?? = pT(pT+1)/2) is one-to-one at ??. We show that using an observable canonical form for the stochastic system under weak conditions ?? is identifiable (d+2, ??). This result is established by explicitly constructing a realization for the non-stationary covariance matrix ??T(??). The standard results [13,14] concerning the realization of stationary covariance matrices follow from our main theorem. The notion of (T,??) identifiability is employed in the strong consistency theorem for maximum likelihood estimators for the parameters of linear time invariant systems from non-stationary cross-sectional data which is to be found in [1].