# pairwise correlation of three random variables

Assume three random variables have all equal pairwise correlation. What are the possible values of this correlation? Can all of these values be achieved?

The solution says $\rho \in [-\frac 12,1]$, but without any explanation. Can someone give me a hint?

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This math.stackexchange.com/questions/59813/… might also be interesting. Not so long ago ... –  Gottfried Helms Dec 27 '12 at 4:17

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 $\rho\in [-\frac{1}{d-1},1]$.

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.

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Thank you for the hint. How do you prove the Lemma you cited? –  John Peter Dec 27 '12 at 3:39
@JohnPeter: I added a brief explanation. –  Yury Dec 27 '12 at 3:59