What guarantees the existence of a finite representation of the Wold decomposition? Mechanics and Intuition

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?

• do you want a nice proof of it or are you asking more about the practicality of the result ? Herman Bierns has the nicest proof of the Wold Decomposition that I have seen. If you google for it, I think it should come up. If not, let me know and I can look. As far as the practical part, every AR(1), can be written as an infinite MA, so that may be connected to the answer. Great question. – mlofton Dec 3 '18 at 17:20
• @mlofton: Thank you for the Bierns reference. I found it but it is far too complex for me... I do not yet understand "sub-Hilbert spaces". I also want to point out that my primary interest is not so much in the Wold representation, which is a beautiful result, but of no practical consequence because we cannot estimate an infinite number of parameters, but rather on the approximation of this infinite Wold representation by a ratio of finite order lag polynomials, which is of enormous practical consequence because we can estimate the parameters of these finite polynomials, hence ARMA. – ColorStatistics Dec 3 '18 at 17:45
• I apologize for getting his name wrong: It's Bierens but you found it anyway. Now I understand your interest better. I forget the name but then check out a paper by Jorgensen. Hold on, I'll try to find it. The idea is that an AR(1) is an infinite MA so that are not as finite as they look. I found it. This is the paper that I think might help you. econometricsociety.org/publications/econometrica/1966/01/01/…. If you have jstor ( I use JPASS, it's pretty reasonable ), you can get it. If you can't, I have it. – mlofton Dec 4 '18 at 8:06
• Keep in mind that I haven't read that paper in a long time so I can't guarantee it will help. But note that a certain specific case of a distributed lag model ( the koyck distributed lag model ) is a specific ARIMAX model. So, they're all kind of related ( ARIMA, ARIMAX, distributed lags etc ) but I don't recall if the paper addresses your question explicitly. Still, it's worth checking out. Sometimes you never know from where the light will enter. – mlofton 6 secs ago edit – mlofton Dec 4 '18 at 8:18
• @ColorStatistics.I truly wasn't sure about it's ulitlity but I'm glad to hear that it sounds helpful. All the best. – mlofton Dec 5 '18 at 16:07