Let (R,m) be a Noetherian local ring I, J two ideals of R and M a finitely generated R-module. Let k ≥ −1 and r _k  = depth _k (I, J ⁿ M/J ⁿ⁺¹ M) be the length of a maximal (J ⁿ M/J ⁿ⁺¹ M)-sequence in dimension >k in I defined by Brodmann and Nhan ( 2008 ). It is first shown that r _k becomes independent of n for large n. Then we prove in this article some sets of associated primes related to regular sequences and the number  r _k in dimension >k for k=-1, 0, 1 are stable for large n. 

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