Let
R
 be a Noetherian local ring, I an ideal of R, and N a finitely generated R-module. Let k≥−1 be an integer and r =\depth_k
(I, N) the length of a maximal N-sequence in dimension > k in I defined by Brodmann and Nhan in (2008). For a subset S ⊆ Spec R, we set S_{≥k} = {p ∈ S | \dim
(R/p)≥k}. We first prove in this article that \Ass_R(
H^j_I(
N)_{≥k} is a finite set for all j≤r. Let  N=⊕_{n≥0}N_n be a finitely generated graded R-module, where R is a finitely generated standard graded algebra over R_0=R. Let r be the eventual value of depth_k(
I,N_n). Then our second result says that for all l≤r the sets \Union_{j≤l}\Ass_R(
H^j_I(
N_n)_{≥k} are stable for large n.

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