# Suppose that the lifetime of mr browns car is

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Unformatted text preview: n N (t)/t ! as t ! 1 (3.1) Proof of Theorem 3.1. We use the Theorem 3.2. Strong law of large numbers. Let x1 , x2 , x3 , . . . be i.i.d. with Exi = µ, and let Sn = x1 + · · · + xn . Then with probability one, Sn /n ! µ as n ! 1 Taking xi = ti , we have Sn = Tn , so Theorem 3.2 implies that with probability one, Tn /n ! µ as n ! 1. Now by deﬁnition, TN (t) t &lt; TN (t)+1 Dividing by N (t), we have TN (t) TN (t)+1 N (t) + 1 t · N (t) N (t) N (t) + 1 N (t) By the strong law of large numbers, the left- and right-hand sides converge to µ. From this it follows that t/N (t) ! µ and hence N (t)/t ! 1/µ. Our next topic is a simple extension of the notion of a renewal process that greatly extends the class of possible applications. We suppose that at the time of the ith renewal we earn a reward ri . The reward ri may depend on the ith interarrival time ti , but we will assume that the pairs (ri , ti ), i = 1, 2, . . . are independent and have the same distribution. Let N (t) R(t) = X i=1 ri 103 3....
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