How to verify that a tentative state-space representation of an ARMA(1,1) process is valid Brockwell & Davis example 9.1.2. show the state-space representation for {$Y_t$}, an ARMA(1,1) process. They claim that the 2 equations below are the observation equation and state equation respectively for {$Y_t$}:
(1) ${Y}_t=[\theta $    $  1] \begin{bmatrix} X_{t-1}\\X_t\end{bmatrix}$
(2) $\begin{bmatrix} X_{t}\\X_{t+1}\end{bmatrix}=\begin{bmatrix} 0 & 1\\0 & \phi  \end{bmatrix}\begin{bmatrix} X_{t-1}\\X_t\end{bmatrix}+\begin{bmatrix} 0 \\Z_{t+1}\end{bmatrix}$
with $\mathbf X_1=\begin{bmatrix} \sum_{j=0}^{\infty}\phi^jZ_{-j} \\\sum_{j=0}^{\infty}\phi^jZ_{1-j}\end{bmatrix}$ and $Z_t$~$WN(0,\sigma^2)$
I have 2 questions here:

*

*How can I convince myself that the above is a valid state-space representation for the ARMA(1,1) process?

*How do I think about what the unobserved state vector $\mathbf X_t$=$\begin{bmatrix} X_{t-1}\\X_{t}\end{bmatrix}$ represents in this case?


Provided that $|\phi| < 1$ we can as in the OP define $X_t
:= \sum_{j = 0}^\infty \phi^j Z_{t -j}$, so $(1-\phi B) \,X_t =
Z_t$. Let us denote temporarily by $Y^\star_t$ the observation of the
candidate State-Space (SS) representation. Since $Y_t^{\star} := \theta
X_{t-1} + X_{t}$, by applying $(1 - \phi B)$ to each side of the last
relation, we get the relation (1) with $Y^\star_t$ replacing
$Y_t$. So, the same model holds for $Y^\star_t$ and $Y_t$. And the SS
representation with $\mathbf{X}_t := [X_{t-1}, \, X_t]'$ gives the
wanted model for $Y_t$.
The series $X_t$ applies a linear filter $1/(1 - \phi B)$ to the white
noise $Z_t$. We can describe $X_t$ as a "coloured noise", yet I can
not see a better interpretation to answer to question 2.  To get an
interpretable (equivalent) representation, we could go with the state
vector $\mathbf{V}_t = [Y_{t},\, \hat{Y}_{t+1|t}]'$ and the
observation equation $Y_t = [1,\, 0] \mathbf{V}_t$. This generalises
to the $\text{ARMA}(p, \,q)$ case with $r := \max\{p,\,q +1\}$ by
taking $\mathbf{V}_t := [Y_{t},\, \hat{Y}_{t+1|t},
\,\dots,\,Y_{t+r-1|t}]'$ with the obvious observation equation $Y_t =
[1,\, 0, \, \dots,\,0]'\mathbf{V}_t$. This is sometimes called the
Akaike SS representation.
Note also that in a SS representation, we need an initial covariance
matrix, be it that $\mathbf{X}_0$ or that of $\mathbf{X}_1$, in both
cases conditional on the empty observation set preceding the
observation of $Y_1$. For the SS representation of the
$\text{ARMA}(1,\,1)$ above, this covariance matrix is easily derived.
