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For a random variable $X \sim f(x)$, the $n$'th moment is defined to be $E_f[X^n]$. But for a multivariate distribution, $X$ is a vector so this definition doesn't make sense since you can't raise a vector to power.

How are moments defined for distributions over vector spaces?

My suspicion is they involve Kronecker products: $$E_f\left[X \otimes X \otimes \cdots \otimes X\right]$$ since the first and second moments of a multivariate Gaussian are $\mathbf{\mu}$ and $\mathbf{\Sigma}$.

Davide Giraudo
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ted
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    You can always define a moment like$$\mathbb{E}[X_1^{n_1}\cdots X_k^{n_k}].$$Check Peter McCullagh's entry on multivariate moments and cumulants on [Scholarpedia](http://www.scholarpedia.org/article/Cumulants). – Xi'an Jan 11 '15 at 08:39
  • Sometimes there is clarity in generality: http://stats.stackexchange.com/q/132914/36229 – shadowtalker Jan 11 '15 at 13:48

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