= Gaussian Gibbs sweep autocorrelation
{c}
{title2=$Z_{2,r}=\rho^2Z_{2,r-1}+\sqrt{1-\rho^2}(\rho\eta_{1,r}+\eta_{2,r})$}
In a systematic <Gibbs sampler> for a standardized <bivariate normal distribution> with <correlation> $\rho$, update $Z_1$ conditionally on the old $Z_2$, then update $Z_2$ using the new $Z_1$. Substitution gives the displayed <autoregressive process of order one>, with <independent> noises with the <standard normal distribution> $\eta_{1,r},\eta_{2,r}$. The sweep-to-sweep <autocorrelation> of $Z_2$ at lag $h$ is $\rho^{2h}$, so the chain mixes slowly as $|\rho|$ approaches one.
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