Convergence of least squares identifiers of time series with martingale difference and binary white Markov generating processes

Abstract

This paper gives some results concerning the convergence of sequential least squares (SLS) identifiers of autoregressive (AR) time series models. Convergence depends on the second moment ergodicity (SME) of the generating (input) process to the time series model. Since SME conditions are difficult to prove for specific generating sequences, this paper proves this condition to be satisfied with respect to two concrete classes of generating sequences, namely, martingale difference sequences and binary white noise Markov sequences to give insight to the convergence of identifiers via binary white noise interrogation inputs. This extends the result of Mann and Wald of convergence for the case of independent and identically distributed (IID) input processes to concrete non-IID input classes.