A probability transition matrix P on a state space S is called doubly stochas- tic if its column sums are all 1, i.e., Σies P(i, j) = 1 for every j Є S. (a) If S is finite and P is doubly stochastic, show that all states of the Markov chain are positive recurrent. (b) If, in addition to the assumptions in part (a), P is also irreducible and aperiodic, then deduce that 1 Pn (i, j) → |S| as n→ ∞. What does this tell you about equilibrium distributions of the Markov chain? (c) If S is countably infinite and P is an irreducible, doubly stochastic transition matrix, show that either all states are null-recurrent or all states are transient. What does this tell you about equilibrium distributions of the Markov chain?