Question: 5.5 Let {X n } be a random walk for which zero is an absorbing state and such that from a positive state, the process

5.5 Let {X_n} be a random walk for which zero is an absorbing state and such that from a positive state, the process is equally likely to go up or down one unit. The transition probability matrix is given by (5.9) with r0 = 1 and p, = q; = ; for i ? 1.

(a) Show that {X_n} is a nonnegative martingale.

(b) Use the maximal inequality II, (5.7) to limit the probability that the process ever gets as high as N > 0.

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