Recall that an $AR(p)$-process (with mean $0$) is defined by the following recurrence relation $$ X_t = \phi_1X_{t-1} + \cdots + \phi_pX_{t-p} + Z_t $$ where $Z_t$ is i.i.d. white noise. If we use the backshift operator $$ B(X_t) = X_{t-1} $$ then we can rewrite this recurrence relation as a polynomial in the operator $B$ $$ \phi(B)X_t = Z_t $$ I keep running into the technical claim that the process is stationary if the roots of $\phi(B)$ (when considered as a complex valued polynomial) lie outside of the complex unit circle $e^{i\theta}$. Why is this true?

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    $\begingroup$ Although I don't fully understand it, I think this is an answer to your question: [When is an “ARIMA process” stationary? ](stats.stackexchange.com/questions/187196/…) $\endgroup$
    – Nayef
    Oct 25, 2019 at 3:36
  • $\begingroup$ The answer here may answer your question (see Kasparis 2006 for more detailed proof). $\endgroup$
    – Ben
    Oct 25, 2019 at 4:15


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