In Section 4.2.4 of [1], the authors write

In this section we consider a causal linear process $$ X_t = \sum_{j = 0}^\infty a_j \varepsilon_{t - j}, \quad t \in \mathbb{N}, $$ where, without loss of generality, $\sum_{j = 0}^\infty a_j^2 = 1$ and $\varepsilon_t$ $(t \in \mathbb{Z})$ are i.i.d. zero mean random variables with Var$(\varepsilon_1) = \sigma^2_\varepsilon < \infty$. Thus, Var$(X_1) = \sigma^2_X = \sigma^2_\varepsilon$. Note that Gaussian processes are included in this definition, but the class is much more general.

I read the last sentence as "all Gaussian processes are causal linear processes". And [2] seems to support that. It argues that *"...the Mallows MA closure is exhausted by three types of processes. The first type is the set of stationary Gaussian processes with mean zero..."*.

However, a counter-example can be constructed by a Gaussian process which is defined as $X(t) = Y$ for all $t \in \mathbb{N}$ where $Y$ is a zero-mean Gaussian random variable. This is an (admittedly degenerate) Gaussian process and I fail to see how it can be expressed as a linear process.

So, is my counter-example wrong? Or do we need additional assumptions for a Gaussian process to be a (causal) linear process? [1] often works with regularly varying auto-covariance functions resp. spectral densities. Has this maybe been assumed implicitly? If so, how do these assumptions help to prove the claim?

# References

[1] J. Beran, Y. Feng, S. Ghosh, and R. Kulik. Long-memory processes. Springer, 2016.