Let S =X, be a random walk and let A,, i > 0, denote the probability that a ladder height equals ithat is, = P{first positive value of S, equals i}.

(a) Show that if 9. j= -1 P{X, = j}= 9 + = 1, j 1, then , satisfies = + 9(+1 +), i > 0.

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(b) If P{X, j, j = -2, -1, 0, 1, 2, show that = 1+5 3+ 5 2 = 3+ 5