Suppose that we have three continuous variables A, B, and C, whose joint probability distribution is P(A,B,C) = P.
We are given that the two covariances
$\langle AB \rangle - \langle A \rangle \langle B \rangle$ and $\langle AC \rangle - \langle A \rangle \langle C \rangle$ are both positive (bounded from below by some q>0).
In the above, $\langle X \rangle$ denotes the average of X with the probability distribution P.
What bounds (if any) does the positivity of the latter two covariances imply on the third covariance $\langle BC \rangle - \langle B \rangle \langle C \rangle$?
* In a related vein, can B and C be independent? *
I must apologize if I overlooked earlier questions (and answers) on this website. There were questions concerning correlations yet these seemed to concern other issues. Nonetheless, I may have easily missed previous pertinent questions.
Write $\langle X Y \rangle - \langle X \rangle \langle Y \rangle = \text{Cov}(X,Y)$, the covariance of $X$ and $Y$, $\sigma(X) = \sqrt{\text{Cov}(X,X)}$ the standard deviation of $X$, and $\rho(X,Y) = \text{Cov}(X,Y)/(\sigma(X) \sigma(Y))$ the Pearson correlation coefficient of $X$ and $Y$. Then it can be shown that $$ \rho(B,C) \ge \rho(A,B) \rho(A,C) - \sqrt{1 - \rho(A,B)^2} \sqrt{1 - \rho(A,C)^2}$$
In particular, if $\rho(A,B) > t>0$ and $\rho(A,C) > t>0$, then $\rho(B,C) > 2 t^2 - 1$.
In order for this to guarantee that $B$ and $C$ are positively correlated (and thus not independent), you'd need $t > 1/\sqrt{2}$.
EDIT: You can obtain the inequality as follows. For convenience, normalize the random variables $A, B, C$ so that their means are $0$ and standard deviations are $1$ (note that transformations $X \to a X + b$ for $a > 0$ do not change correlation coefficients). Let $B_1 = B - \rho(A,B) A$. Then $\langle B_1 A \rangle = 0$ and $\langle B_1 \rangle = 0$ so $\sigma(B_1)^2 = \langle B_1^2 \rangle = 1 - \rho(A,B)^2$. Similarly if $C_1 = C - \rho(A,C) C$, $\sigma(C_1)^2 = 1 - \rho(A,C)^2$. And now using Cauchy-Schwarz, $$ \eqalign{ \rho(B,C) &= \langle (B_1 + \rho(A,B) A)( C_1 + \rho(A,C) A) \rangle\cr &= \langle B_1 C_1 \rangle + \rho(A,B) \rho(A,C) \cr &\ge \rho(A,B) \rho(A,C) - \sigma(B_1) \sigma(C_1) \cr &= \rho(A,B) \rho(A,C) - \sqrt{1 - \rho(A,B)^2} \sqrt{1 - \rho(A,C)^2}}$$