Here we obtain results on the covariance matrices of the random vectors of unconditional convex bodies. We prove that a number of sample points proportional to the dimension is enough to reconscruct the covariance matrix with given error. This improves on previously known results, and can be sen as a weak version of Bai-Yin theorem (the strong version for l_p balls was obtained in the previous paper).

In a recent paper by Adamczak, Litvak, Pajor and Tomczak-Jaegermann, the result has been extended to all convex bodies (not necessarily unconditional).

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aubrun (arrobas) math. univ-lyon1. fr