I am trying to make the most of computations that have already been performed in previous steps of an algorithm. Throughout this problem statement I am only mentioning correlation, but I think it is essentially an equivalent question if I replace "correlation" with "covariance".

Suppose I have samples of three random variables $X_1$, $X_2$, and $X_3$ (each a column vector of dimensions $(n, 1)$). If I have already computed 2 correlations $c_{1, 2} = cor(X_1, X_2)$ and $c_{1, 3} = cor(X_1, X_3)$, what can I say about the remaining correlation $c_{2, 3} = cor(X_2, X_3)$? It seems like I should be able to place upper and lower bounds $c_{2, 3}$.

One possible example is that if there is high positive correlation between $X_1$ and $X_2$ and also between $X_1$ and $X_3$, then I'd like to show that there must be a reasonably strong positive correlation between $X_2$ and $X_3$ (by computing the bounds ($b_1$ and $b_2$):

$c_{1, 2} = 0.99$

$c_{1, 3} = 0.99$

$b_1 \le c_{2, 3} \le b_2$

The information I have available about $X_1$, $X_2$ and $X_3$ is there sum and sum of squares: $\sum_j x_{i, j} \\ \\ i \in {1, 2, 3}$ $\\ \\ \\ $ $\sum_j x_{i, j}^2 \\ \\ i \in {1, 2, 3}$. Based on those values I can calculate the mean and variance of $X_1, X_2, X_3$.

I also have the sum of the products of the pairs $X_1, X_2$ and $X_1, X_3$: $\sum_j x_{1, j} \cdot x_{2, j}$ $\\ \\ \\ $ $\sum_j x_{1, j} \cdot x_{3, j}$. Based on those values and the means and variances I can calculate $c_{1, 2}, c_{1, 3}$

Specifically I do not have: $\sum_j x_{2, j} \cdot x_{3, j}$, which would allow me to calculate $c_{2, 3}$.

I use this equation to calculate the correlations (from http://en.wikipedia.org/wiki/Correlation_and_dependence>Wikipedia):

$cor(a, b) = \dfrac{n\sum {a_i b_i} - \sum {a_i} \sum {b_i}}{\sqrt{n\sum{a_i^2} - (\sum{a_i})^2}\sqrt{n\sum{b_i^2} - (\sum {b_i})^2}}$

Thanks in advance for your insight!