# Does uncorrelation imply independence for marginally Gaussian random variables?

Let $X_1, \ldots, X_n$ be marginally Gaussian distributed random variables that are uncorrelated. Does it imply that they are independent?

• There's an example of dependent uncorrelated random normals here. It's possible to use the same basic idea to generate others. SIlverfish's answer here gives another. – Glen_b -Reinstate Monica Sep 9 '16 at 0:46

If $X = (X_1,\ldots, X_n)$ are jointly normal, too, then yes. Otherwise, no.
In this case $\Sigma = \text{diag}(\sigma_1^2,\ldots, \sigma_2^2)$ and $\mu = (\mu_1,\ldots,\mu_n)'$
For an example of two independent $X_1$ and $X_2$ that are uncorrelated, but dependent, check out the example here. You can take $n=2$. Define $X_1 \sim \text{Normal}(0,1)$, $W$ is $1$ or $-1$ with probability $.5$ and independent from $X_1$. Then define $X_2 = WX_1$.
The $X$s are un-correlated because \begin{align*} \text{Cov}(X_1,X_2) &= \text{Cov}(X_1,W X_1)\\ &= E[X_1^2W] \\ &= E[X_1^2]E[W] \\ &=0 \end{align*}