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Assume a random vector $$X=(X_1,\ldots,X_p) \sim N(0,\Sigma)$$ with $\Sigma$ positive definite matrix. Is there literature regarding the covariance (formulas) of the form $E[X_j^kX_{j'}^l]$? with $k,l \in \{1,\ldots, q\}$ for some $q\in \mathbb{N}$.

Additional Note: More concretely i am concerned whether a covariance matrix of the form $\tilde{\Sigma} = \left( E[X_j^kX_{j'}^l] \right)_{l,k, j,j'}$ is still positive definite

statuser123
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For any couple $(j,j')$, it suffices to calculate the characteristic function of $(X_j,X_{j'})$: $$\phi_{(X_j,X_{j'})}(t_1,t_2)=\mathbb{E}\left( e^{t_1 X_j +t_2X_{j'}} \right) \tag{1}$$ The formula of $(1)$ can be easily deduce from this.

We notice that $$ X_j^k X_{j'}^l \cdot e^{t_1 X_j +t_2X_{j'}} = \frac{\partial ^{(k+l)}}{\partial^kt_1\cdot \partial^l t_2} \left( e^{t_1 X_j +t_2X_{j'}}\right)$$

Then, the expectation $\mathbb{E}\left( X_j^k X_{j'}^l \right) $ can be calculated from the derivatives $(k+l)$ of the characteristic function at the point $(t_1,t_2)=(0,0)$: $$\color{red}{\mathbb{E}\left( X_j^k X_{j'}^l \right) = \frac{\partial ^{(k+l)}\phi_{(X_j,X_{j'})}}{\partial^kt_1\cdot \partial^l t_2}(0,0)} $$

NN2
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    Thank you very much! That helped a lot. – statuser123 Mar 06 '24 at 16:44
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    @statuser123 You're welcome. And if you need to calculate many $\mathbb{E}\left( X_j^k X_{j'}^l \right)$, then it's better to compute the characteristic function of $(X_1,...,X_p)$ (denoted by $\phi_{(X_1,...,X_p)}(t_1,t_2,...,t_p)$ and then $$\mathbb{E}\left( X_j^k X_{j'}^l \right) = \frac{\partial ^{(k+l)}\phi_{(X_1,...,X_p)}}{\partial^kt_1\cdot \partial^l t_2}(0,0,...,0) $$ – NN2 Mar 06 '24 at 17:00