Covariance matrix of uniform distribution in $L^p$ Euclidean ball

Question

Let $$X=\operatorname{Unif}\left(\left\{x:\ \sum_{i=1}^n |x_i|^p\le 1\right\}\right)$$

Is there some method to calculate the covariance matrix of $X$?

Thank you!

Answer 1

If $\xi_i,\ldots,\xi_n$ are i.i.d. random variables with density $f(x)=(2\Gamma(1+1/p))^{-1}e^{-|x|^p}$ and $\zeta\sim \text{exp(1)}$ independent of $\xi\equiv (\xi_i,\ldots,\xi_n)^{\top}$, then $$ X=\frac{\xi}{(\sum_{i=1}^n |\xi_i|^p+\zeta)^{1/p}} $$ is uniformly distributed in the unit $L_p$-ball (ref). Since $x\mapsto xf(x)/(|x|^p+w)^{1/p}$, $w\ge 0$, is an odd function, $\mathsf{E}X_i=0$. Similarly, $\mathsf{E}X_iX_j=0$ for $i\ne j$. Finally, the second moment of $X_i$ is $$ \mathsf{E}X_i^2=\int_0^\infty \int_{\mathbb{R}^n}\frac{x_i^2}{(\sum_{k=1}^n |x_k|^p+z)^{2/p}}\prod_{k=1}^n f(x_k)\times e^{-z}\,dx\,dz, $$ which can be computed at least numerically.