Let $X$, $X'$, and $X''$ be i.i.d. random vectors taking values in $\mathbb{R}^N$. Is it true that
$$\text{cov}(\lVert X-X' \rVert, \lVert X-X'' \rVert) \geq 0?$$
My numerical simulations suggest that it is, but I have so far only been able to show the following:
- $\text{cov}(\lVert X \rVert, \lVert X \rVert) = \text{var} (\lVert X \rVert) \geq 0$
- $\text{cov}(\lVert -X' \rVert, \lVert -X'' \rVert) = \text{cov}(\lVert X' \rVert, \lVert X'' \rVert) = 0$
- $\text{cov}(X-X',X-X'') = \text{var}(X) \geq 0$ for $X, X', X''$ with dimension $1$