I know that there are many example of correlated normal random variables which are not jointly (multivariate) normal. However, are there conditions which state when correlated normal random variables are jointly 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.

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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).

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