Minimum value of constant correlation

Question

Given a set of random variables $X_1,\dots,X_n$ such that $${\rm cov}(X_i,X_j)=\rho$$ for $i\neq j$. What is the minimum value of $\rho$ possible? Can it be negative? Thanks!

Answer 1

For ease, suppose we normalize so that all random variables have mean $0$ and variance $1$. What you are asking is equivalent to for what values of $\rho$ is the supposed covariance matrix $\Sigma_{\rho}$ with diagonal $1$ and off-diagonals $\rho$ positive semi-definite. This is possible precisely when $\rho\in [\frac{-1}{n-1},1]$; when $\rho=\frac{-1}{n-1}$, one can easily see that $\Sigma_{\rho}$ is then diagonally dominant, and therefore p.s.d. Moreover, $\Sigma_{\rho}$ is easily seen to be p.s.d. when $\rho=1$, as this is easily seen to be rank-one with nonnegative eigenvalue (a more probabilistic argument is this is the all-ones matrix, and this is realizable as a covariance matrix by taking $X_1=\ldots=X_n$). For all other $\rho$ in that range, one may take convex combinations to show positive semi-definiteness.

For $\rho<\frac{-1}{n-1}$, simply observe $(1,\ldots,1)^T$ is an eigenvector with strictly negative eigenvalue.