Gaussian two-coordinate Gibbs recursion ($Y_{n+1}={\rho }^2{Y}_n+\sqrt{1-{\rho }^2}(\rho Z_1+Z_2)$)

For a standard bivariate normal distribution with correlation $\rho$, each conditional has variance $1-{\rho }^2$ and mean $\rho$ times the other coordinate. A sweep draws $X_{n+1}=\rho Y_n+\sqrt{1-{\rho }^2}{Z}_1$, then $Y_{n+1}=\rho X_{n+1}+\sqrt{1-{\rho }^2}{Z}_2$. The resulting coordinate chain is an AR(1) process with coefficient ${\rho }^2$ and innovation variance $1-{\rho }^4$. It shows explicitly why near-unit correlation slows successive Gibbs sweeps.