# Are random variables correlated if and only if their ranks are correlated?

Assume $X,Y$ are continuous random variables with finite second moments. The population version of Spearman's rank correlation coefficient $\rho_s$ can be defined as the Pearson's product-moment coefficient $\rho$ of the probability integrals transforms $F_X(X)$ and $F_Y(Y)$, where $F_X,F_Y$ are the cdf's of X and Y, i.e.,

$\rho_s(X,Y) = \rho(F(X),F(Y))$.

I wonder whether one can generally conclude that $\rho(X,Y) \neq 0 \leftrightarrow \rho(F(X),F(Y))\neq 0$? I.e., do we have linear correlation if and only if we have linear correlation between the ranks?

(Note: If $X$ and $Y$ are positively quadrant dependent, i.e., $\delta(x,y) =F_{X,Y}(x,y)-F_X(x)F_y(Y)> 0$ then Hoeffding's covariance formula $Cov(X,Y) = \int\int\delta(x,y)dxdy$ yields that $\rho(X,Y)>0$ and $\rho(F(X),F(Y))>0$. However, this is only a special case and I want to know whether the equivalence also holds for general dependence structures)

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