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If $X_1,\ldots,X_n$ are dependent normal random variables, what would be the distribution of $X_1+\ldots+Xn$? Is it still normal?

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It depends on how they are dependent.

The answer is yes if they are multivariate normal but not always in general.

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    @May: if they are independent (as in the first *i* of *iid*) then $\sum_i \sum_j X_i Y_j = \sum_i X_i \times \sum_j Y_j $. So if they are normally distributed then you have the product of two normal random variables, which is in general [not normally distributed](http://mathworld.wolfram.com/NormalProductDistribution.html)2012-11-16