On learning about ARMA(p,q) models, Box and Jenkins (1970) defined a very important class of stochastic processes that is obtained as a white noise process goes through a linear filter.
This can be written as:
$\ w_t = \sum_{k=0}^{\infty} \Psi_k . a_{t-k}$, [1]
where$\ w_t$ is the stochastic process, $\Psi_k$ is a linear filter, and $\ a_{t-k}$ is a white noise process.
Box and Jenkins defined the linear filter as
$\Psi_k = {\large \frac{\theta_q(B)}{\phi_p(B)}}$, [2]
where $\theta_q(.)$ and $\phi_p(.)$ are polynomials of the type $\ P(x) = 1 - c_1.x - c_2.x^2 - ... - c_k.x^k$, $\ B$ is the lag operator and a variable of the both aforementioned polynomials.
One can write [1] as
$\ w_t = \theta_q(B) . \phi^{-1}(B).a_t$ [3]
About equation [3], I read:
It's evident that when $\phi_p(B)$ is inverted to obtain $\ w_t$ as a function of $\ a_t$, there will be a polynomial up to an infinite degree multiplying $a_t$ since $\phi_p(B)$ is a finite polynomial.
My question is: Is there a law that says that an inverse of a finite-degree polynomial will always be an inifinite-degree polynomial?
I have looked for such proof, but so far to no avail. Thank you for your feedback.