I know that if a random $n$-dimensional variable $Y= [Y_, ..., Y_n]^T$ has density $f$, then its covariance matrix $\Sigma $ is positive definite: $\forall x \in \mathbb{R}^n \setminus \{0\}: x^T \Sigma x >0$.
I wonder if the reverse implication is true.
$\Sigma>0 \ \implies Y$ has a density function.
I've tried to find such a theorem, but I failed. I do not know whether it is false or very trivial.
Could you help me with that?
It appears that this is not true for $n-$dimensional random variables in general.
Will it help if we assume that $Y$ has an $n-$dimensional normal distribution?