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Proof: D's updates go:
i.e. D is simply the average of all samples so far.
As , .
Proof: D's updates go:
As :
that is, .
One way of looking at this
is to consider as the average of all samples before time t,
samples which are now irrelevant for some reason.
We can consider them as samples from a different distribution f:
Hence:
as .
I could have made this clearer:
An alternative approach (e.g. see [Grefenstette, 1992]) is to build up not the average of all samples, but a timeweighted sum of them, giving more weight to the most recent ones. This is accomplished by setting to be constant, in which case D's updates go:
Since as , this weights more recent samples higher. Grefenstette uses typically .
Consider sampling alternately from two stationary random variables d and f:
Proof: From Theorem A.1, after 2t samples:
as .
More generally,
consider where we have multiple stationary distributions ,
where each distribution has expected value .
At each time t, we take a sample from one of the distributions.
Each distribution has probability p(i) of being picked.
Repeat:
Proof: For a large number of samples t we expect to take p(1) t samples from , p(2) t samples from , and so on. From Theorem A.1, we expect:
as .
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