When does depth stabilize early on?

In this paper we study graded ideals I in a polynomial ring S   such that the numerical function k↦depth(S/Ik)k↦depth(S/Ik) is constant. We show that, if (i) the Rees algebra of I is Cohen–Macaulay, (ii) the cohomological dimension of I   is not larger than the projective dimension of S/IS/I and (iii) the K-algebra generated by some homogeneous generators of I is a direct summand of S  , then depth(S/Ik)depth(S/Ik) is constant. All the ideals with constant depth-function discovered by Herzog and Vladoiu in [11] satisfy the criterion given above. In the contest of square-free monomial ideals, there is a chance that a converse of the previous fact holds true.