The depth of the associated graded ring of ideals with any reduction number

Let R be a local Cohen–Macaulay ring, let I be an R-ideal, and let be the associated graded ring of I. We give an estimate for the depth of when is not necessarily Cohen–Macaulay. We assume that I is either equimultiple, or has analytic deviation one, but we do not have any restriction on the reduction number. We also give a general estimate for the depth of involving the first powers of I, where denotes the Castelnuovo regularity of and ℓ denotes the analytic spread of I.