Gaussian Gibbs sweep autocorrelation ($Z_{2,r}={\rho }^2{Z}_{2,r-1}+\sqrt{1-{\rho }^2}(\rho {\eta }_{1,r}+{\eta }_{2,r})$)

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.