TITLE: Is correlation (in some sense) transitive?
QUESTION [14 upvotes]: If we know that A has some correlation with B ($\rho_{AB}$), and that B has some with C ($\rho_{BC}$), is there something we know to say about the correlation between A and C ($\rho_{AC}$)?
Thanks.

REPLY [23 votes]: Correlations are cosines of angles in $L^2$ hence, if $\varrho_{AB}\geqslant 0$ and $\varrho_{BC}\geqslant 0$, then
$$
\varrho_{AC}\geqslant\varrho_{AB}\varrho_{BC}-\sqrt{1-\varrho_{AB}^2}\cdot\sqrt{1-\varrho_{BC}^2}.
$$
Thus, if $\varrho_{AB}\geqslant c$ and $\varrho_{BC}\geqslant c$ with $c\geqslant0$, then
$$
\varrho_{AC}\geqslant2c^2-1.
$$
For example, if $\varrho_{AB}\geqslant90\%$ and $\varrho_{BC}\geqslant90\%$ then $\varrho_{AC}\geqslant62\%$.