A strong correlation between A and B. Strong correlation between B and C. No correlation between A and C. Is this possible?

Suppose that the correlation ρABρAB between AA and BB and the correlation ρBCρBC between BB andCC are both the same number ρρ. Then ρACρAC lies between 2ρ2−12ρ2−1 and 11, the light blue shaded region below.

If they’re fully correlated, ρ=1ρ=1, then, of course, ρACρAC is also 1. But as ρρ decreases from 1, the minimum value for ρACρAC decreases four times as fast. If ρ=0.9ρ=0.9, then ρACρAC could be almost as low as 0.60.6. With ρ=0.7ρ=0.7, AA and BB could be uncorrelated.

The only constraint on a correlation matrix, knowing nothing else is that |R|≥0|R|≥0. For three variables with correlations s,t,r this leads to the inequality

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