14. Wald’s equation can be used as the basis of a proof of the elementary renewal theorem. Let X1,X2, . . . denote the interarrival times of a renewal process and let N(t) be the number of renewals by time t.

(a) Show that whereas N(t) is not a stopping time, N(t) + 1 is.

Hint: Note that N(t) = n ⇔ X1 + · · · + Xn t and X1 + · · · + Xn+1 > t

(b) Argue that

(c) Suppose that the Xi are bounded random variables. That is, suppose there is a constant M such that P{Xi

(d) Use the previous parts to prove the elementary renewal theorem when the interarrival times are bounded.

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N(t)+1 E x] = mlm(0) + 1] =[m(t) i=1