Is there a general equation or procedure for computing higher-order co-moment matrices (i.e., coskewness matrix, cokurtosis matrix, etc) for a vector of random variables?

For example, the covariance matrix of a vector of random variables $\mathbf{X} = (X_1, ..., X_N)$ is an $N \times N$ matrix $\mathbf{C}(\mathbf{X})$ defined by

$$ \mathbf{C}(\mathbf{X}) = \mathrm{E}\big[(\mathbf{X}-\mathrm{E}[\mathbf{X}])(\mathbf{X}-\mathrm{E}[\mathbf{X}])^\top \big] $$

Which can be written equivalently as:

$$ \mathbf{C}(\mathbf{X}) = \mathrm{E}[\mathbf{X}\mathbf{X}^\top]-\mathrm{E}[\mathbf{X}]\mathrm{E}[\mathbf{X}]^\top $$


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