A Class of Non‐Embeddable ARMA Processes

Abstract

We show that a stationary ARMA(p, q) process {Xn = 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 {Xn}. 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).