There are n states and for each pair of states i and j, a positive number d ij = d ji is given. A particle moves from state to state in the following manner: Given that the particle is in any state i,...

There are n states and for each pair of states i and j, a positive number d_ij
= d_ji
is given. A particle moves from state to state in the following manner: Given that the particle is in any state i, it will next move to any j ≠ i with probability P_ij
given by

Assume that P_ii
= 0 for all i. Show that the sequence of positions is a reversible Markov chain and find the limiting probabilities.