Is correlation (in some sense) transitive?

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.

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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 $\varrho_{AC}\geqslant62\%$.

I am afraid this is a non-answer: probably.

But imagine tossing a fair coin $100$ times. Let $A$ be the number of heads in the first $50$ tosses, $B$ the number of heads in the full $100$ tosses, and $C$ the number of heads in the last $50$ tosses.

Then $A$ and $B$ are (weakly) positively correlated, as are $B$ and $C$, but $A$ and $C$ have correlation $0$.

By working a little harder, we could even get negative correlation between $A$ and $C$.

This is a small ($2$ instead of $100$) version of André’s answer, so as to make correlations easier to compute. Flip two coins. Let $A$ be $1$ if the first coin comes up heads, and $0$ if tails. Define $C$ similarly using the second coin. Let $B$ be $1$ if both coins come up heads, and $0$ otherwise.