# When are correlated Normal random variables multivariate Normal? [duplicate]

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.

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 $$f_{X,Y}(x,y) = \begin{cases} 2\phi(x)\phi(y), & x \geq 0, y \geq 0,\\ 2\phi(x)\phi(y), & x < 0, y < 0,\\ 0, &\text{otherwise},\end{cases}$$ 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).

It certainly is possible.

From a theoretical perspective, there are many different ways to "characterize" the Multivariate Normal distribution, see for example Hamedani, G. G. (1992). Bivariate and multivariate normal characterizations: a brief survey. Communications in Statistics-Theory and Methods, 21(9), 2665-2688.

From a practical perspective see for example Henze, N. (2002). Invariant tests for multivariate normality: a critical review. Statistical papers, 43(4), 467-506.

• Thank you for those references. I am looking in Section 4 for sufficient conditions which only rely on the marginals. If you know of any useful theorems, I would appreciate the suggestion. – EliK Oct 17 '18 at 19:25
• @EliK Since both an MVN joint distribution and a non-MVN distribution can readily be constructed to have identical margins (e.g. via the use of a non-Gaussian copula and transforming each margin to normality), there can be no sufficient conditions which rely only on the margins. – Glen_b Oct 18 '18 at 0:49
• I got that impression from reading through the paper. Thank you for pointing that out explicitly. – EliK Oct 18 '18 at 19:37

This is an interesting question. I will look at it from another viewpoint: When should you expect that a joint distribution with normal marginals, not is multinormal?

Certain phenomena occurring in data cannot be described by a multinormal distribution, and some examples (even a list) of such phenomena is interesting. Two examples, as a start: If the random vector $$(Y,X^T)^T$$ is multinormal (take here $$Y$$ as scalar), then the conditional expectation of $$Y$$ given $$X$$ takes the form of a linear function in the $$X$$: $$\DeclareMathOperator{\E}{\mathbb{E}} \E [Y \mid X=x]= \beta_0 + \beta^T x$$ for some parameters $$\beta_0, \beta$$ (which can be calculated form the expectation and covariance matrix of $$(Y,X^T)^T$$). So, if data indicates that regressing $$Y$$ on $$X$$ is nonlinear, or needs interaction terms, then the joint distribution cannot be multinormal. For an example see Conditional expectation of two identical marginal normal random variables.

Another example is Can I analyze or model a conditional correlation? which is about studying the correlation between two variables conditional on a third, how the correlation changes with values of the third. If the three variables are multinormal, one can show easily that the conditional correlation is a constant, so this phenomenon cannot occur.

But there must be many other interesting such examples ...