Summary: G. Weiss [J. Appl. Probab. 12, 831-836 (1975; Zbl 0322.60037)] has shown that for causal autoregressive moving-average (ARMA) models with independent and identically distributed (i.i.d.) noise, time- reversibility is essentially unique to Gaussian processes. This result extends to quite general linear processes and the extension can be used to deduce that a non-Gaussian fractionally integrated ARMA process has at most one representation as a moving average of i.i.d. random variables with finite variance.
In the proof of this uniqueness result, we use a time-reversibility argument to show that the innovations sequence ( one-step prediction residuals) of an ARMA process driven by i.i.d. non-Gaussian noise is typically not independent, a result of interest in deconvolution problems. Further, we consider the case of an ARMA process to which independent noise is added. Using a time-reversibility argument we show that the innovations of the ARMA process with added independent noise are independent if and only if both the driving noise of the process and the added noise are Gaussian.