Show that if a Markov chain is periodic with period greater than 2, then the chain is not…

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Show that if a Markov chain is periodic with period greater than 2, then the chain is not reversible. For the random walk in Example 94, specify its periodicity for the following cases: (a) ri = 0 for each i. (b) ri > 0 for some i.

Example 94

Random Walk. Consider a random walk Xn on S = {0, 1,…} with transition probabilities

Assume the pi and qi are positive. Then clearly this Markov chain is irreducible. Now, its detailed balance equations (1.62) (for the two respective cases j = i + 1, and j = i − 1 ) are