  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.
