11. For the random walk of Example 4.13 use the strong law of large numbers to give another proof that the Markov chain is transient when Hint: Note that the state at time ç can be written as £?= x Yt where the 17s are independent and P{Yt = lj = ñ = 1 - P{Yt = -1}. Argue that if

ñ > y, then, by the strong law of large numbers, Óº Y{\-> oo as ç -> oo and hence the initial state 0 can be visited only finitely often, and hence must be transient. A similar argument holds when ñ < j .