# Multivariate Gaussian fitting

When trying to approximate a distribution of random vectors $$D$$ by using multivariate Gaussian, what properties must we ensure that $$D$$ has? I.e., what distributions can be estimated by multivariate Gaussian fitting?

In a multivariate Gaussian distribution different coordinates can be related only in a linear way. Run regression of $$X(i)$$ on $$X(j)$$ and $$X(j)^2$$. If the coefficient of $$X(j)^2$$ is statistically significant, then the multivariate distribution is not Gaussian. A scatterplot matrix allows one to detect any aberrations informally.
Also, it goes without saying that the marginal distribution of each coordinate $$X(i)$$ must be "bell-shaped". In other words, it should pass Gaussian goodness-of-fit tests. Such tests are Kolmogorov-Smirnov, Shapiro-Wilk, Jarque-Bera, ...
Multivariate Gaussian distribution is not allowed to be degenerate. In particular, the covariance matrix of the coordinates must have positive determinant. For example, $$Y$$ can be a Gaussian variable but $$(Y, -Y)$$ will not be a Gaussian vector since the coordinates are linearly related.