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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 – Henry Nov 16 '12 at 0:52

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