We consider a symmetric random walk in two dimensions whose state space is the set {(i, j ) : i — 0,1,2; j == 0,1,2}. Moreover, we suppose that the boundaries are reflecting. That is, when the process...

We consider a
symmetric
random walk in two dimensions whose state space is the set {(i, j ) : i — 0,1,2; j == 0,1,2}. Moreover, we suppose that the boundaries are
reflecting.
That is, when the process makes a transition

that would take it outside the region defined by the state space, then it returns to the last position it occupied (on the boundary).

(a) Calculate the one-step transition probability matrix of the Markov chain.

(b) Show that the limiting probabilities exist and calculate them.