Perfectly random sampling of truncated multinormal distributions

The target measure $\mu$ is the distribution of a random vector in a box $\text{cB}$, a Cartesian product of bounded intervals. The Gibbs sampler is a Markov chain with invariant measure $\mu$. A ``coupling from the past’’ construction of the Gibbs sampler is used to show ergodicity of the dynamics and to perfectly simulate $\mu$. An algorithm to sample vectors with multinormal distribution truncated to $\text{cB}$ is then implemented.