What is the distribution of sum of dependent normal random variables?

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?

• @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