 Solution

For a target probability density function $\pi ({\theta }_1,\dots ,{\theta }_p)$, one systematic sweep of the Gibbs sampler updates coordinates successively:Each update leaves the target invariant: keeping the other coordinates with their target marginal and replacing one coordinate by its conditional distribution reconstructs the target joint probability density function. Therefore composition of the coordinate updates also preserves $\pi$. It defines a Markov chain whose consecutive draws are generally dependent. Under irreducibility, aperiodicity and the usual recurrence conditions it converges to the target law from an arbitrary valid start; a stationary start gives exact target marginals immediately. Invariance alone does not prove these ergodicity conditions, and a systematic full sweep need not itself be reversible.