Autocovariance tail recurrence of a causal ARMA process (${\gamma }_k-{\sum }_{j=1}^p{\varphi }_j{\gamma }_{k-j}=0(k>q)$)

For a causal autoregressive moving-average model driven by independent white noise, multiply its recursion by $X_{t-k}$ and take covariance. When $k>q$, all driving white noise values in its moving-average term occur strictly after $t-k$, so their covariances with $X_{t-k}$ vanish. The resulting homogeneous tail recurrence has the same characteristic roots as the autoregressive model recursion. Distinct roots of modulus below one give exponential autocovariance decay.