For a stationary $AR(p)$ model, $\theta(B)X_t = w_t$, how can I show that $\theta(z) \neq 0$ for all $|z| = 1$.

I tried it as:

$\theta(z) = (1-\frac{1}{\lambda_1}z)(1-\frac{1}{\lambda_2}z)...(1-\frac{1}{\lambda_p}z)$

where $\lambda_1, \lambda_2, …\lambda_p$ are the roots of the polynomial $\theta(z)$.

The inversion of each term is defined for $|\frac{1}{\lambda_i}| < 1; \forall i$ and $|z| \leq 1$. But I am not sure how to proceed it further from here.


Let $\theta(1)=0$. This means that $1$ is a root of $\theta(B)$. $1$ is not outside the unit circle, which means $\theta(B)X_t = w_t$ is not a stationary AR model (see Hamilton 1995, p. 58). This contradicts the assumption.


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