12. Wald's equation can be used as the basis of a proof of the elementary renewal theorem. Let Xx, X2,... denote the interarrivai 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) = ç ï Xx + ··· + Xn < t and Xx + ··· + Xn+X > t

(b) Argue that N(t)+l Ó Xi é = ]
= ßlm(t) + 1]

(c) Suppose that the Xt are bounded random variables. That is, suppose there is a constant M such that P{Xi < M) = 1. Argue that t< Ó Xi

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