A Class of Non‐Embeddable ARMA Processes

We show that a stationary ARMA( p , q ) process { X n = 0, 1, 2, ...} whose moving‐average polynomial has a root on the unit circle cannot be embedded in any continuous‐time autoregressive moving‐average (ARMA) process { Y }( t ), t ≥ 0}, i.e. we show that it is impossible to find a continuous‐time ARMA process { Y }( t )} whose autocovariance function at integer lags coincides with that of { X n }. This provides an answer to the previously unresolved question raised in the papers of Chan and Tong ( J. Time Ser. Anal. 8 (1987), 277–81), He and Wang ( J. Time Ser. Anal. 10 (1989), 315–23) and Brockwell ( J. Time Ser. Anal. 16 (1995), 451–60).