Hello -- I need to know if such a theorem exists.
Suppose I have a sequence of RVs: X1, X2, ...
each with mean mu and finite second momment.
Now, I know that I have asymptotic normality of
the standardized sample mean if the RVs are independent.
The rate of the convergence is SQRT(n).
What I want to know is: is there a theorem that states
asymptotic normality for the standardized mean at a
rate n^b for some b? In particular, does it allow for a
relaxing of the independence assumption? For example,
if there was a dependence within the sequence, perhaps asymptotic
normality be achieved with an n^(3/4) rate.
Any thoughts or references are appreciated.
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