Sum of $n$ correlated normal random variables

Question

I can only find results on sums containing two normal random variables, for example here https://stats.stackexchange.com/questions/19948/what-is-the-distribution-of-the-sum-of-non-i-i-d-gaussian-variates. How does the $n$ variable sum version look?

Answer 1

Use induction.

E.g. if the distributions of $X_1, X_2, X_3$ and their covariances are given, set $Y_1 = X_1 + X_2$ and compute its distribution.

Then compute $cov(Y_1,X_3) = cov(X_1,X_3) + cov(X_2,X_3)$.

Then compute the distribution of $Y_1 + X_3 = X_1 + X_2 + X_3$, and so on.

Answer 2

If the random variables $X_1,X_2,\ldots,X_n$ are each marginally normally distributed, then depending upon the joint distribution the sum $\sum_{j=1}^n X_j$ may or may not be normally distributed.

If $\mathbf{X} = (X_1,X_2, \ldots , X_n)'$ has a (joint) multivariate normal distribution with mean $\mathbf{\mu}$ and covariance matrix $\mathbf{\Sigma}$, then for any nonsingular matrix $\mathbf{C}$, the random vector $\mathbf{Y} =\mathbf{C} \mathbf{X}$ has a multivariate normal distribution with mean $\mathbf{C}\mathbf{\mu}$ and covariance matrix $\mathbf{C} \mathbf{\Sigma} \mathbf{C}'$.

Taking $\mathbf{C}$ where each element in the first row equals $1$, we have $Y_1 = \sum_{j=1}^n X_j$, and this random variable does in fact have a normal distribution.