Is it true that \(\operatorname{Cov}\left(\left|X-X^{\prime}ight|,\left|Y-Y^{\prime}ight|ight)=0\) implies that \(X\) and \(Y\) are independent? (Recall that \(X, X^{\prime}\) are iid; \(Y, Y^{\prime}\) are iid.)

It is not hard to show that the following example is a counterexample. Define the two-dimensional pdf

\[
p(x, y):=(1 / 4-q(x) q(y)) I_{[-1,1]^{2}}(x, y)
\]

with

\[
q(x):=-(c / 2) I_{[-1.0]}+(1 / 2) I_{(0, c)}
\]

where \(c:=\sqrt{2}-1\).