# Wide-sense stationary linear process = ARMA process?

In Brockwell and Davis' Introduction to Time Series and Forecasting,

a linear process is defined to be

An ARMA process/model is defined to be

Note that by "stationary", the book means "wide-sense stationary".

An ARMA process can also be written as a form of linear process

1. Is it correct that a stochastic process is an ARMA process, if and only if it is a linear process and also wide-sense stationary?
2. Despite that the definition of an ARMA process says $X_t$ only depends on past values of $X$ and $Z$, is it correct that an ARMA process may not be causal, because the solution to (3.1.1) is (3.1.3)? Note a causal ARMA process is defined in (3.1.5) here. Do I misunderstand something?

Thanks and regards!

• Notice that in a ARMA process, $X_t$ is a function of $Z_t$ and $q$ past $Z_i$ (and $p$ past $X_i$) but no future $Z_i$ (that is, $Z_{t+1}, Z_{t+2}$, etc.) while a linear process has no such restrictions. These kinds of questions are dealt with (both in the stochastic sense here as well as for deterministic sequences) in the signal processing literature, and asking a similar question over on dsp.SE might bring forth useful answers too. – Dilip Sarwate Jul 26 '13 at 12:34
• @DilipSarwate: Thanks! But the definition of ARMA process allows it to depend on future values of $Z_i$. See books.google.com/…. Am I wrong? – Tim Jul 26 '13 at 12:41
• I am looking at the definition (3.1.1) that you have supplied which does not allow $X_t$ to depend on future values of $Z_i$ (or future $X_i$ either for that matter). If you have a different definition in mind, either edit the current question to include it, or ask a different question. – Dilip Sarwate Jul 26 '13 at 12:45
• @DilipSarwate: (1) An ARMA process isn't necessarily causal, because the solution to (3.1.1) is (3.1.3) which shows that an ARMA process may not be causal. (2) A causal ARMA process is defined in (3.1.5) in the link in my last comment, where the definition of an ARMA process is still (3.1.1). Do I misunderstand something? – Tim Jul 26 '13 at 12:56
• @DilipSarwate: I have edited my post to include my confusion in the comments. – Tim Jul 26 '13 at 15:32

An ARMA process corresponds to a rational transfer function. The definition of linear process given is more general, as it allows transfer functions that are not rational.

• Can you provide a reference for this claim? – denizen of the north Aug 10 '18 at 17:46