17. For the random walk of Example 4.18 use the strong law of large numbers to give another proof that the Markov chain is transient when p = 12

.

Hint: Note that the state at time n can be written as

ni

=1Yi where the Yis are independent and P{Yi = 1} = p = 1 − P{Yi = −1}. Argue that if p > 12

, then, by the strong law of large numbers,

n1 Yi →∞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 < 12

.