= Two-sided stationary inverse of an autoregressive polynomial
{title2=$A(B)X_t=\epsilon_t$}
If an <autoregressive model> has <white noise> of positive <variance>, a unique <weakly stationary process> solving its equation on all integer times exists precisely when its <autoregressive polynomial> has no root on the <unit circle>. In that case $1/A(z)$ has an absolutely summable <Laurent series> on an annulus containing the <unit circle>; its coefficients define a bilateral <linear filter of a stationary time series>. Applying that inverse also proves uniqueness among bounded second-moment solutions. A root on the <unit circle> forces the candidate <spectral density of a stationary process> to have a nonintegrable singularity. The stronger <causality root criterion for an autoregressive model> requires every root outside the closed <unit disk>.
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