On convergence of least-squares identifiers of autoregressive models having stable and unstable roots

A proof is presented for establishing the convergence of least-squares (LS) identification algorithms when applied to autoregressive (AR) time-series models where some or all poles my be unstable, i.e., outside the unit circle in the complex
z
-plane. The only assumption on the time-series model is that its residual or driving sequence is a zero-mean uncorrelated (white noise) sequence with finite second moment which is second-moment-ergodic (SME). In cases where the SME condition cannot be established, the resulting identified parameters will relate to a model driven by an SME process which is the LS approximation to the actual process whose identification was sought.