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The following paper: "Concentration of norms and eigenvalues of matrices" by M. Meckes (who is activeactive on MO) seems answer the question. In particular, theorem 1 of that paper proves an inequality of the type requested in the comments above (it proves something more general, but I have not checked if all the conditions in the OP hold).

The following paper: "Concentration of norms and eigenvalues of matrices" by M. Meckes (who is active on MO) seems answer the question. In particular, theorem 1 of that paper proves an inequality of the type requested in the comments above (it proves something more general, but I have not checked if all the conditions in the OP hold).

The following paper: "Concentration of norms and eigenvalues of matrices" by M. Meckes (who is active on MO) seems answer the question. In particular, theorem 1 of that paper proves an inequality of the type requested in the comments above (it proves something more general, but I have not checked if all the conditions in the OP hold).

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Suvrit
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The following paper: "Concentration of norms and eigenvalues of matrices" by M. Meckes (who is active on MO) seems answer the question. In particular, theorem 1 of that paper proves an inequality of the type requested in the comments above (it proves something more general, but I have not checked if all the conditions in the OP hold).