Autocovariance of an AR(1) process observed with white noise
= Autocovariance of an AR(1) process observed with white noise
{title2=$\gamma_Y(h)=\frac{\sigma_z^2\phi^{|h|}}{1-\phi^2}+\sigma_w^2\mathbf1_{\{h=0\}}$}
For a causal AR(1) plus uncorrelated observation white noise, cross covariances vanish by L2 convergence of the autoregressive noise expansion. Its covariance is $\sigma_z^2\phi^{|h|}/(1-\phi^2)+\sigma_w^2\mathbf1_{\{h=0\}}$. The additional noise changes only lag zero, attenuating the normalized positive-lag correlations.