Causality root criterion for an autoregressive model ($\Phi (z)\ne 0(|z|\le 1)$)

A causal time series satisfying $\Phi (B)X_t={\epsilon }_t$ with nondegenerate white noise has a square-summable present-and-past filter. The identity $\Phi (z)\Psi (z)=1$ excludes roots inside the unit disk; a pole of $1/\Phi$ on its boundary also prevents square summability. Thus all roots of $\Phi$ lie strictly outside the unit circle.

The inverse autoregressive polynomial is an analytic function on a disk larger than the unit disk. The Cauchy estimate gives exponentially decaying coefficients of its infinite moving-average representation. Summing their products gives $|\gamma (h)|\le C{s}^{|h|}$ for some $s\in (0,1)$.