For which x the matrix above is the covariance matrix?
\begin{pmatrix} 1 & x & x \\ x & 1 & x \\ x & x & 1 \\ \end{pmatrix}
I only understand it must be positively-definite, but how to prove that all these matrices are realizable with some vector $(\xi_1, \xi_2, \xi_3)$?
If it is positive definite and symmetric then it can be diagonalised by a rotation in 3-space so it is sufficient to prove that all diagonal matrices with positive eigenvalues are realizable. But that can easily be achieved using 3 independent variables with the appropriate variances.
The solution to the original problem is then obtained by rotating the 3 variables back, which gives linear combinations of them.
Thus it is sufficient to find all $x$ for which the matrix is positive definite.