Consider the one-dimensional symmetric random walk of Example 4.18, which was shown in that example to be recurrent. Let πi denote the long-run proportion of time that the chain is in state i.
(a) Argue that πi = π0 for all i. (b) Show that ? πi ̸= 1.
i
(c) Conclude that this Markov chain is null recurrent, and thus all
πi = 0.
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