A transition probability matrix P is said to be doubly stochastic if the sum over each column equals one; that is, P = 1, for all j If such a chain is irreducible and aperiodic and consists of M + 1 states 0, 1, ..., M, show that the limiting probabilities are given by M+ j = 0, 1, ..., M *15. A particle moves on a circle through points which have been marked 0, 1, 2, 3, 4 (in a clockwise order). At each step it has a probability p of moving to the right (clockwise) and 1 p to the left (counterclockwise). Let X, denote its location on the circle after the nth step. The process (X, n 0) is a Markov chain.

(a) Find the transition probability matrix.

(b) Calculate the limiting probabilities.