(Solved): Let N2 be an integer. We can consider {0,1,,N1} to be a "circle" by assuming that N1 is ...

Let $N\ge 2$ be an integer. We can consider $\{0,1,\cdots ,N-1\}$ to be a "circle" by assuming that $N-1$ is adjacent to 0 as well $N-2$. Let $X_n$ be simple random walk on the circle. The transition probabilities are $p_{k,k-1}=p_{k-1,k}=0.5,k=1,\cdots ,N-1,p_{0,N-1}=p_{N-1,0}=0.5$. Let $N=6$, (a) (10 points) What is the transition matrix $\mathbf{P}$ ? (b) (10 points) Is there a limiting probability vector? If yes, what is it? If no, what is the period? (c) (10 points) Is there any invariant probability distribution? If yes, what is it? Is it unique? If no, why?