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Let $X_i \in L_2$ be a sequence of pairwise correlated random variables. The random variables can have arbitrary positive correlation but can't have arbitrary negative correlation.

How can I show that for $(X_1,\ldots,X_n)$ and $\forall i,j \in \{1,\ldots,n\}, i\neq j$

$$\mathrm{Cor}(X_i,X_j)<\frac{-1}{n-1}$$ is not possible

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  • $\begingroup$ Are you sure that your claim is correct ? What if $n = 2$, $X_1 = -X_2$ with variance equal to $1$ ? $\endgroup$ – dallonsi Dec 11 '18 at 15:27
  • $\begingroup$ @dallonsi, it is correct; note the strict inequality in the question, whereas your example has an equality. $\endgroup$ – Marcus M Dec 11 '18 at 17:21
  • $\begingroup$ yes Marcus, thanks. My mistake $\endgroup$ – dallonsi Dec 13 '18 at 13:11
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The idea is to realize that covariance matrices are positive semi-definite. Define variables $Y_j = X_j/ \sqrt{\operatorname{Var}(X_j)}.$ Seeking a contradiction, suppose that the condition you mention holds, i.e. $$\operatorname{Cor}(X_i,X_j) = \operatorname{Cov}(Y_i,Y_j) < -\frac{1}{n-1}.$$

Let $\Sigma$ be the covariance matrix of $(Y_1,\ldots,Y_n)$. Then $\Sigma$ has $1$'s on the diagonal, and each off-diagonal entry is less than $-\frac{1}{n-1}$. If we define $x = (1,1,\ldots,1)^T$, then we see that $x^T \Sigma x < 0$, which contradicts the fact that covariance matrices are positive semi-definite.

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