We know that the Wold Decomposition Theorem says that any purely nondeterministic covariance-stationary process, $x = [x_t : t \in \mathbb{Z}]$, can be written as a linear combination of lagged values of a white noise process:
$$x_t = \sum_{j=0}^{\infty}\psi_j u_{t-j}$$
According to this lesson lesson (slides 12-14), under general conditions, the infinite lag polynomial of the Wold decomposition can be approximated by the ratio of two finite-lag polynomials:
$$\psi(L) \approx \frac{\theta(L)}{\phi(L)}$$
in other words, the process $x$ can be accurately approximated by a ARMA process in the following sense: $$x_t^* = \frac{\theta(L)}{\phi(L)} u_t $$
I would like to understand better this approximation.
- Is this approach via sequences of ARMA processes?
- Does the approximation use any metric?
- How can we determine the $\phi(L)$ and $\theta(L)$ polynomials? Are they unique?
Do you have some reference (book, paper etc) or insight?