If I understand correctly, for various versions of the central limit theorems (CLT), when applying to a sequence of random variables, each random variable is required to have finite mean and finite variance, plus some other conditions depending on the version of the CLT.
Months ago, I heard of something, probably about some case when the classical CLT fails, which I haven't been able to understand. I am not sure if my following description is correct, but that is perhaps the best I can recall:
if one random variable in the sequence dominates (in some sense, such as in terms of magnitude?) the other random variables, then the central limit theorem doesn't hold.
I was wondering if someone is able to figure out what the quote is trying to say?
Thanks and regards!
PS: A paper named Asymptotic Distribution Theory for the Kalman Filter State Estimator was mentioned regarding the above quote. I don't quite understand the paper, so cannot figure out how it helps to clarify the quote. But I guess Section "3.2 Remarks on Theorems and Corollaries" on page "1999" might be related.