Wold Representation for an ARMA (1,1) I have this ARMA(1,1) process where $\epsilon_t$ is the classical White Noise process
$$X_t=\epsilon_t +\alpha_{t-1}\epsilon_{t-1}+\theta_{t-1}X_{t-1}$$ 
and I have to write its Wold representation. Using the lag operator I get 
$$\epsilon_t=\frac{1-\theta_{t-1}L}{1+\alpha_{t-1}L}X_{t}$$ Assuming the process is stationary and invertible, how can I recover the Wold representation?
 A: The Wold's theorem states that every covariance-stationary time series $X_T$ can be written as the sum of two time series, one deterministic and one stochastic. What we are searching for is something of the following form:
$$X_t = \sum_{t=0}^{\infty} \gamma^j \varepsilon_{t - j} +\eta_t .
$$
How can we achieve this if we are given the process $y_t = \phi y_{t-1} -\theta \varepsilon_{t-1} +\varepsilon_t?$ By substitution! Write
\begin{align*}
y_{t} =& \phi( \phi y_{t-2} - \theta\varepsilon_{t-2}+\varepsilon_{t-1}) - \theta \varepsilon_{t-1}+\varepsilon_t,\\
=& \phi^2 y_{t-2} - \phi\theta \varepsilon_{t-2} + \phi\varepsilon_{t-1} - \theta \varepsilon_{t-1} +\varepsilon_t,\\
=&\phi^2( \phi y_{t-3} - \theta\varepsilon_{t-3}+\varepsilon_{t-2}) - \phi\theta \varepsilon_{t-2} + \phi\varepsilon_{t-1} - \theta \varepsilon_{t-1} +\varepsilon_t,\\
\vdots&
\end{align*}
We see that a pattern emerges and that this process can be written as an infinite sum of error terms:
$$ y_t =  \sum_{j=1}^{\infty}\phi^{j-1} ( \phi - \theta)\varepsilon_{t-j} + \varepsilon_t.
$$
That is, we have rewritten the ARMA(1,1) as an MA($\infty$) process.
