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. Simple Random Walk on a Circle. Let N 2 2 be an integer. We can consider {0, 1, - -- ,N 1} to be a \"circle\" by assuming that N 1 is adjacent to U as well N 2. Let X\" be simple random walk on the circle. The transition probabilities are PMs1 pic1,}: = 051k = 11' ' ' 3N _11p0,N1 = pN1,0 = 0H5 Let N = 63 (a) (5 points) What is the transition matrix P? (b) (5 points) Is there a limiting probability vector? If yes, what is it? If no, what is the period? c 5 points Is there an invariant probability distribution? If yes, what is it? Is it 3' unique? If no, why? d 5 points Is there any recurrent state? If es, what is it? If no, why? 3' e 5 points Is there any transient state? If es, what is it? If no, why? 3