This question arose from the following observation.
It is well known that if a random vector $(X, Y)$ is bivariate normal and $X, Y$ are uncorrelated, then $X, Y$ are independent. However, if we drop the assumption that $(X, Y)$ is bivariate normal, then there exist uncorrelated but not independent random variables $X, Y$. Even if we add an assumption that the marginals $X, Y$ are normal, such an example exists. Therefore, it seems that the jointly normal assumption is a critical assumption to deduce independence.
On the other hand, most known counterexamples seem not have a joint density. Then can we replace the joint normal assumption with a joint density assumption?
In precise, if $X, Y$ are normal random variable with a joint density and are uncorrelated, then can we conclude $X, Y$ are independent?
I think if we can show that the assumptions imply joint normality, then we can reduce the claim to the known fact. But, this does not seem trivial either.
Any help is highly appreciated!