# Higher order moments of a multivariate Gaussian rv

Let $$X~N_d(\mu,\Sigma)$$ be a multivariate Gaussian random vector. Is there a convenient formula for each of $$\mu_p\triangleq \mathbb{E}\left[\sum_{i=1}^d |X_i|^p\right],$$ in terms of $$\mu$$ and of $$\Sigma$$?

Reasoning This is true for the univariate case; see this table. For example, when $$d=1$$ and $$p=3$$, then $$\mu_p= \mu^3 + 3\mu\sigma^2$$.

This is to be distinguished from this question, which consideres the second higher-order moment but defining it via the outer and not inner product.

• "or example, when d=1 and p=, then" .... did you intend to up a 3 in there after "p="? Mar 13, 2019 at 3:59

Since $$\mu_p\triangleq \mathbb{E}_{\mu,\sigma}\left[\sum_{i=1}^d |X_i|^p\right]=\sum_{i=1}^d\mathbb{E}_{\mu,\sigma}\left[|X_i|^p\right]$$ the multivariate nature of $$X=(X_1,\ldots,X_n)$$ and in particular the correlations between the $$X_i$$'s have no relevance. For a Normal distribution $$\mathcal{N}(\mu,\sigma^2)$$, the moments are connected by the recurrence relation (Patel & Read, 1986)$$\xi_p\triangleq\mathbb{E}_{\mu,\sigma}[X^p]=\mu\xi_{p-1}+(p-1)\sigma^2\xi_{p-2}$$which provides $$\mu_{2q}=\xi_{2q}$$. As stated by Wikipedia]2, the generic absolute moment of order $$p>0$$ is $$\mathbb{E}_{\mu,\sigma}\left[|X|^p \right] =\sigma^p 2^{p/2} \frac {\Gamma\left(\frac{1+p} 2\right)}{\sqrt\pi} {}_1F_1\left( -\frac{p}{2}, \frac{1}{2}, -\frac{1}{2} \left( \frac \mu \sigma \right)^2 \right)$$