This is not true in general. Consider two independent fair die rolls, with A and C being the outcomes. Let B be the sum of the outcomes. The higher A is, the h

This is not true in general. Consider two independent fair die rolls, with A and C being the outcomes. Let B be the sum of the outcomes. The higher A is, the higher B is likely to be, so A and B are positively correlated. So are B and C. However there is no correlation between A and C as they are independent.