Solved – When are correlated Normal random variables multivariate Normal?

Say I observe n univariate random variables $X_1, dots, X_n$ that are each $N(mu, sigma^2)$ with common correlation $rho$. Is it possible that these are jointly normal? If so, what are the conditions and how would I know if they are jointly normal.

There are no conditions based only on the marginal pdfs that can ensure joint normality. Let $phi(cdot)$ denote the standard normal density. Then, if $X$ and $Y$ have joint pdf $$f_{X,Y}(x,y) = begin{cases} 2phi(x)phi(y), & x geq 0, y geq 0,\ 2phi(x)phi(y), & x < 0, y < 0,\ 0, &text{otherwise},end{cases}$$ then $X$ and $Y$ are (positively) correlated standard normal random variables (work out the marginal densities to verify this if it is not immediately obvious) that do not have a bivariate joint normal density. So, given only that $X$ and $Y$ are correlated standard normal random variables, how can we tell whether $X$ and $Y$ have the joint pdf shown above or the bivariate joint normal density with the same correlation coefficient ?

In the opposite direction, if $X$ and $Y$ are independent random variables (note the utter lack of mention of normality of $X$ and $Y$) and $X+Y$ is normal, then $X$ and $Y$ are normal random variables (Feller, Chapter XV.8, Theorem 1).