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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 righthand
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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 Spring '10
 DURRETT
 The Land

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