The multivariate normal distribution seem to not follow the central limit theorem since the random variables are correlated (not independent).
Can someone explain how they can be normally distributed? (though it’s something we will deal with at level 2)
The Central Limit Theorem applies to a single random variable.
Of course it wouldn’t apply to a multivariate distribution. By definition.
There is a multivariate CLT (but presumably beyond the scope of CFA Level I).
For me to clear this confusion, can you please briefly conceptually explain how multivariate follows CLT although the variable are dependent (which contradict with concept of CLT, that is independent variables)?
Not entirely sure - but I believe your intuition is correct. Even with the multivariate CLT, the random variables must be i.i.d.
Multivariate distributions are not a topic of discussion on the Level I exam, at least that I’ve seen.
The correct statement should be: The sum of i.i.d multivariate variables, will follow the multivariate normal distribution according to the multivariate CLT.
The CLT is a particular case of multivariate CLT when the dimension of iid multivariate variable is equal to 1. And you can not use the CLT for multivariate variables.
PS: I think this concept is out of scope of CFA program. But if you want to understand more about the multivariate CLT, you should google the three definitions: “multivariate random variable”, “iid” and “multivariate Central Limit Theorem”.