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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!

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closed as off-topic by Brian Borchers, dantopa, Leucippus, mrtaurho, Javi Mar 25 at 11:29

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  • $\begingroup$ How do you define the uniform distribution on such a set? It is usually only defined on sets of finite measure. Here, I don't even see a measure. $\endgroup$ – amsmath Mar 24 at 3:10
  • $\begingroup$ I forget to add $|\cdot|$. For example, when $p=2$, the set is a ball. For $p\rightarrow\infty$, the set is a hypecube. $\endgroup$ – zbh2047 Mar 24 at 3:14
  • $\begingroup$ What have you tried so far? Can you work this out for a very simple case (e.g. the $L_{\infty}$ hypercube for $n=2$)? $\endgroup$ – Brian Borchers Mar 24 at 4:41
  • $\begingroup$ For $L_\infty$ it is easy because $X_i$ are independent. But when $X_i$ are dependent I have no idea to deal with such case. $\endgroup$ – zbh2047 Mar 24 at 5:07
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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.

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