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Let $(R,\m)$ be a Noetherian local ring $I, J$ two ideals of $R$ and $M$ afinitely generated $R-$module. It is first shown that for $k\geq -1$ theinteger $r_k = \depth_k(I,J^nM/J^{n+1}M)$, it is the length of a maximal$(J^nM/J^{n+1}M)-$sequence in dimension $>k$ in $I$ defined by M. Brodmann andL. T. Nhan \cite{BN}, becomes for large $n$ independent of $n$. Then we provein 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 $\bigcup_{j\ler_1}\Ass_R(H^j_I(J^nM/J^{n+1}M))\cup\{\m\}$ are stable for large $n$. We alsoobtain similar results for modules $M/J^nM$.

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