In the book Introduction to Time Series and Forecasting by Peter J. Brockwell and Richard A. Davis, at page 75, there is the Existence and Uniqueness of stationary solution of an ARMA process:
Consider an ARMA$(p,q)$ process: $\phi(B)X_t = \theta(B)Z_t$. A stationary solution $\{X_t\}$ exists (and is also the unique stationary solution) if and only if $$ \phi(z) = 1 - \phi_1 z - \phi_2 z^2 - \dots - \phi_pz^p \neq 0, \quad \forall |z| = 1. $$
The author doesn't provide proof, so here is my attempt to prove the sufficient condition:
Let $z_1,z_2,\dots, z_p$ are $p$ complex roots of polynomial $\phi(z)$ and WLOG, assume $|z_p| = 1$. We are going to prove that $\phi(B)X_t = \theta(B)Z_t$ doesn't exists stationary solution. Suppose the contrary, let $w_j = z_j^{-1}$ for all $j\in\{1,2,\dots, p\}$ and let $\varphi(z) = (1 - w_1z)\dots(1-w_{p-1}z)$. Rewrite the given ARMA process as $$ (1 - w_1B)\dots(1-w_{p-1}B)(1-w_{p}B)X_t = \theta(B) Z_t $$ or $$ \varphi(B) X_t = w_pX_{t-1} + \theta(B) Z_t $$ Apply this equation recursively, we have $$ \varphi(B)^{n+1} X_t = w_p^{n+1}X_{t-1} + \sum_{j=0}^n \varphi(B)^j\theta(B) Z_t, \quad\forall n\in\mathbb{N} $$ Hence $$ \left| \varphi(B)^{n+1} X_t - w_p^{n+1}X_{t-1}\right| = \left| \sum_{j=0}^n \varphi(B)^j\theta(B) Z_t\right| $$ Here is my idea, we are going to prove that $\mathrm{Var}[\mathrm{R.H.S}] \ge C_1n\sigma_Z^2$ for some constant $C_1$ independent from $n$, and $\mathrm{Var}[\mathrm{L.H.S}] \leq C_2\gamma_X(0)$ for some constant $C_2$ independent from $n$. Finally, let $n\to+\infty$ to get the contradiction. But it seems like I cannot find $C_2$, so for now I'm stuck.
Any help would be appreciated!