Constraints on correlation coefficients of multiple random variables

Question

I am looking for a generalization of Correlation between three variables question for more than three variables. It is stated in one of the answers there that, for three variables with pairwise correlation coefficients $a$, $b$, and $c$, the following should hold:

$$a \geq bc - \sqrt{1-b^2}\sqrt{1-c^2}.$$

For $N>3$ variables, the above should hold as well for each triple, but the question is whether considering all triples is sufficient to conclude that all $N(N-1)/2$ correlation coefficients make sense as a whole. If not, what else should hold?

My goal is not to check correctness (which can be done by, for example, examining the eigenvalues of the corresponding matrix) but to find constraints for future generative purposes.

Thank you!

Best wishes, Ivan

Answer 1

Let us consider angles $$ \varphi_{ij}=\arccos r_{ij}. $$ It is the great-circle distance between points $i$ and $j$ on the unit $N$-dimensional sphere.

Then for any $k$ points $i_1,i_2,\ldots,i_k$ you have to satisfy inequality

$$ \varphi_{i_1i_k} \le \sum_{l=1}^{k-1} \varphi_{i_li_{l+1}}. $$

Answer 2

This is an answer not to the initial part of your question, but to the "find constraints for future generative purposes".

A correlation matrix must be positive-definite and have unit diagonal. Any correlation matrix $R$ can be generated from a covariance matrix $V$ by $$R = \mathrm{diag}(V)^{-1/2}\, V \,\mathrm{diag}(V)^{-1/2}$$ where $\mathrm{diag}(V)$ is the diagonal matrix obtained from $V$.

The only constraint on a covariance matrix is that it be positive-definite. Any positive-definite matrix $V$ can be written uniquely by Cholesky decomposition as $$ V = L^{\mathrm{T}} L $$ where $L$ is upper-triangular and has positive components on the diagonal. These positive components can in turn be uniquely written as the exponentials of real numbers.

So, to generate a covariance and correlation matrices without worrying about constraints you can do as follows:

1. choose an arbitrary upper-triangular matrix $A$;
2. replace the diagonal components of $A$ with their exponential, obtaining a (uniquely determined) matrix $L$;
3. take the product $L^{\mathrm{T}} L := V$.

The matrix $V$ is a covariance matrix in one-one correspondence with $L$.

Further you can obtain a correlation matrix $R$ by the first formula above. The correspondence $L \mapsto R$ is many-to-one though.

I hope this helps!