Let Z0, Z1, Z2,... be a realization of an ergodic chain. If we sample every kth epoch, then show

(a) that the sampled chain Z0, Zk, Z2k,...

is ergodic,

(b) that it possesses the same equilibrium distribution as the original chain, and

(c) that it is reversible if the original chain is.

Thus, we can estimate theoretical means by sample averages using only every kth epoch of the original chain.