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In the usual causal stationary convention, a centered autoregressive moving-average model satisfiesSet $\Phi (z)=1-{\sum }_{j=1}^p{\varphi }_j{z}^j$ and $\Theta (z)=1+{\sum }_{j=1}^q{\theta }_j{z}^j$. After removal of common factors, the usual causality condition is that all zeros of $\Phi$ lie outside the closed unit disk. Then $\Theta (z)/\Phi (z)$ has an absolutely summable causal coefficient sequence, giving a linear-filter representation of $X$ in terms of white noise. Invertibility of the moving-average part is a separate issue and is not needed for its covariance spectrum.

White noise has one-sided density ${\sigma }^2/\pi$. Applying the filtering result, or equating the spectra of the two filtered sides of the defining equation, givesThe moving-average sign agrees with the positive $\theta$ convention used later. More generally, the same quotient applies to a stationary two-sided solution when the reduced denominator has no unit-circle zeros; causality selects the past-dependent realization. This is the spectral density of an ARMA process.