Asymptotic stability of certain sets of associated prime ideals of local cohomology modules

Let $(R,\m)$ be a Noetherian local ring $I, J$ two ideals of $R$ and $M$ a finitely generated $R-$module. It is first shown that for $k\geq -1$ the integer $r_k = \depth_k(I,J^nM/J^{n+1}M)$, it is the length of a maximal $(J^nM/J^{n+1}M)-$sequence in dimension $>k$ in $I$ defined by M. Brodmann and L. T. Nhan \cite{BN}, becomes for large $n$ independent of $n$. Then we prove in this paper that the sets $\bigcup_{j\le r_k}\Ass_R(H^j_I(J^nM/J^{n+1}M))$ with $k=-1$ or $k=0$, and $\bigcup_{j\le r_1}\Ass_R(H^j_I(J^nM/J^{n+1}M))\cup\{\m\}$ are stable for large $n$. We also obtain similar results for modules $M/J^nM$.