Every covariance stationary process can be written as a linear, infinite distributed lag of white noise. In other words, every covariance stationary process has a Wold representation. Then we go on to say that this infinite distributed lag of white noise can always be approximated by the ratio of 2 finite-order lag polynomials. In other words for every Wold representation (infinite) there is an approximation (finite). It is difficult to overestimate the importance of the existence of this approximation, as without it there would be no ARMA modelling, which is the core of linear time series modelling, and yet every single textbook I’ve seen only mentions the existence of such an approximation in one sentence as if it were a self-evident fact.
(1) Why is it the case that the infinite Wold representation can always be approximated by the ratio of two finite order polynomials? What guarantees the existence of such an approximation?
(2) How good is this approximation? Is the approximation better in some cases than in other?