# 83 using the idea of renewal reward processes we can

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Unformatted text preview: 1. LAWS OF LARGE NUMBERS be the total amount of rewards earned by time t. The main result about renewal reward processes is the following strong law of large numbers. Theorem 3.3. With probability one, R(t) Eri ! t Eti (3.2) Proof. Multiplying and dividing by N (t), we have 0 1 N (t) 1 R(t) @ 1 X A N (t) = ! Eri · ri t N (t) i=1 t Eti where in the last step we have used Theorem 3.1 and applied the strong law of large numbers to the sequence ri . Here and in what follows we are ignoring rewards earned in the interval [TN (t) , t]. These do not e↵ect the limit but proving this is not trivial. Intuitively, (3.2) can be written as reward/time = expected reward/cycle expected time/cycle an equation that can be “proved” by pretending the words on the right-hand side are numbers and then canceling the “expected” and “1/cycle” that appear in numerator and denominator. The last calculation is not given to convince you that Theorem 3.3 is correct but to help you remember the result. A second approach to this is that...
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