Question

Consider the one-dimensional symmetric random walk of Example 4.18, which was shown in that example to...

  1. 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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