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Let D be updated by:
where d is bounded by , , and the initial value of . Then:
Proof: The highest D can be is if it is always updated with :
so . Similarly .
I should note this only works if α is between 0 and 1.
Proof: In the discrete case, Q is updated by:
so by Theorem B.1:
This can also be viewed in terms of temporal discounting:
Similarly:
For example, if , then . And (assuming ) as , .
Note that since , it follows that .
Proof: In the discrete case, W is updated by:
so by Theorem B.1:
by Theorem B.2.
Similarly:
Note that since , it follows that .
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