Here's the thing :) Let $X\sim N_n(\vec{0}, \Sigma)$ be a multivariate normal random variable (centered) of dimension $n\geq 1$. Here, $\Sigma$ denotes the covariance matrix.
My question is: Is it possible to compute (or bound) the directional centered absolute moments? I.e.
$$\int_{\mathbb{R}^n} |x_1|^{p_1}\cdots |x_n|^{p_n} p_{\Sigma}(x_1,\dots,x_n)dx_1\cdots dx_n,$$ where $$p_{\Sigma}(x_1,\dots,x_n) = (2\pi)^{-n/2} |\Sigma|^{-1/2} \exp\left\{ -\frac{1}{2} x^t \Sigma^{-1} x\right\}$$ denotes the density function and $|\Sigma|$ the determinant of $\Sigma$ and $p_1,\dots,p_n$ are positive real numbers.
Thank you very much for the help!