The Metropolis acceptance mechanism (9.6) ordinarily implies aperiodicity of the underlying Markov chain. Show that if the proposal distribution is symmetric and if some state i has a neighboring state j such that πi > πj , then the period of state i is 1, and the chain, if irreducible, is aperiodic. For a counterexample, assign probability

πi = 1 4 to each vertex i of a square. If the two vertices adjacent to a given vertex i are each proposed with probability 1 2 , then show that all proposed steps are accepted by the Metropolis criterion and that the chain is periodic with period 2.