17. For the random walk of Example 4.15 use the strong law of large numbers to give another proof that the Markov chain is transient when p = 1 2 .

Hint: Note that the state at time n can be written as n i=1Yi where the Yis are independent and P{Yi = 1} = p = 1 − P{Yi = −1}. Argue that if p > 1 2 , then, by the strong law of large numbers, n 1Yi → ∞ as n → ∞ and hence the initial state 0 can be visited only finitely often, and hence must be transient.

A similar argument holds when p < 1 2 .