pairwise correlation of three random variables

Hint: There exist random variables $X_1$, $X_2$ and $X_3$ with pairwise correlations $\rho_{12}$, $\rho_{13}$ and $\rho_{13}$ if and only if the matrix $$\Sigma = \begin{pmatrix} 1 & \rho_{12} & \rho_{13}\\ \rho_{12} & 1 &\rho_{23} \\ \rho_{13} & \rho_{23} & 1 \end{pmatrix} $$ is positive semidefinite. We get that there exist random variables $X_1$, $X_2$ and $X_3$ with all pairwise correlations equal to to $\rho$ if and only if the matrix $$\begin{pmatrix} 1 & \rho & \rho\\ \rho & 1 &\rho \\ \rho & \rho & 1 \end{pmatrix} $$ is positive semidefinite (see below for the explanation). This matrix is positive semidefinite precisely when $\rho\in[-1/2,1]$.

Note that there exist $d$ random variables $X_1,\dots,X_d$ s.t. $\mathrm{cor}(X_i,X_j) = \rho$ (for $i\neq j$) if and only if $\rho\in [-\frac{1}{d-1},1]$.

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Here is a brief explanation why there exist random variables $X_1$, $X_2$ and $X_3$ with pairwise correlations $\rho_{12}$, $\rho_{13}$ and $\rho_{13}$ if and only if $\Sigma$ is positive semidefinite.

Suppose that the matrix $\Sigma$ is positive semidefinite. Let $$\Gamma = \begin{pmatrix}\gamma_1 \\ \gamma_2 \\ \gamma_3\end{pmatrix} \sim {\cal N}(0,\Sigma)$$ be the multivariate normal random variable with covariance matrix $\Sigma$. Then the correlation of $\gamma_i$ and $\gamma_j$ equals $\rho_{ij}$.

Now suppose that $X_1$, $X_2$ and $X_3$ has correlations $\rho_{ij}$. We may assume that ${\mathbb E}[X_i] = 0$ and $\mathrm{cov}[X_i]=1$. Then the covariance matrix of $(X_1,X_2,X_3)$ is $$\Sigma = \begin{pmatrix} 1 & \rho_{12} & \rho_{13}\\ \rho_{12} & 1 &\rho_{23} \\ \rho_{13} & \rho_{23} & 1 \end{pmatrix}. $$ Therefore, $\Sigma$ is positive semidefinite.