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Let $N≥2$ be an integer. We can consider ${0,1,⋯,N−1}$ 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 $p_{k,k−1}=p_{k−1,k}=0.5,k=1,⋯,N−1,p_{0,N−1}=p_{N−1,0}=0.5$. Let $N=6$, (a) (10 points) What is the transition matrix $P$ ? (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?

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