Suppose I have $k$ random variables $X_1,\ldots,X_k$ and I want to form their covariance matrix, which is the $k\times k$ matrix where the $(i,j)$th entry is $Cov(X_i,X_j)$. What kind of bounds exists on the possible values in this matrix, or better to say what kind of bounds exists among these variables? I know that it is symmetric and that the diagonals are all nonnegative. But perhaps given $Cov(X_1,X_2)$, $Cov(X_2,X_3)$, $Var(X_1)$, $Var(X_2)$, and $Var(X_3)$, there are new bounds on what $Cov(X_1,X_3)$ can be. Is there a general statement that captures all these new bounds that could occur?

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    $\begingroup$ Yes, a necessary and sufficient condition is that the matrix is symmetric and positive semidefinite, that is, $\Sigma^T=\Sigma$ and $y^T\Sigma y\ge0$ for all vectors $y\in\mathbb R^k$. $\endgroup$ – Rahul Jun 1 '17 at 2:27

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