1. * (Reversibility of a Chain). A stationary chain fXng with transition probabilities pij is called reversible if there is a function m.x/ such that pij m.i/ D pji m.j/ for all i; j 2 S . Give a simple sufficient condition in terms of the function m that ensures that a reversible chain has a proper stationary distri- bution. Then, identify the stationary distribution.
2. Give a physical interpretation for the property of reversibility of a Markov chain.
3. (Reversibility). Give examples of a Markov chain that is reversible and one that is not.