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We present short and elementary proofs of two theorems of Huckaba and Marley, while generalizing them at the same time to the case of a module. The theorems concern a characterization of the depth of the associated graded ring of a Cohen-Macaulay module, with respect to a Hilbert filtration, in terms of the Hilbert coefficient e 1 . As an application, we derive bounds on the higher Hilbert coefficient e i in terms of e 0 .doi:10.1081/agb-120028790 fatcat:lgvpvoli3rbntjdjysevduopca