Is $E[(X_iX_i')^2]$ positive definite if $\text{Var}(X_i)$ is positive definite? Let $X_i \in \mathbb{R}^k$ be a random variable such that $E[(X_iX_i')^2]$ exists and is finite. Suppose further that $\text{Var}(X_i) = E[X_iX_i'] - E[X_i]E[X_i]'$ is positive definite.
It seems clear that $E[X_iX_i']^2$ is positive definite, but is it true that $E[(X_iX_i')^2]$ is necessarily positive definite?

I'm going to discard the index of $X$. 
Definition
$$M\in  \mathbb{R} ^{m\times m} \text{ is positiv semi-definie } \iff  \forall u \in \mathbb{R} ^m , uMu^T \ge 0$$
So
$$C = E[(X X^T)^2]$$
$$u^T C u =  E\big[\big(u^TX X^T\big) \big(X X^T\big)^Tu\big] = E\big[||u^T X X^T ||_2^2\big] \ge 0$$
$E[X X^T]^2$ is not necessary positive definite. For example: $X=0$. For positive definitness you need each eigenvalue to be positive, which means that apart from positive semi-definite property (which is shown above). You need the $C$ matrix to be non-singular. So
$$\det C = \det E[(X X^T)^2] > 0 \iff \det C \ne 0$$
