stability of the depth function of good filtrations

Let a be a noetherian local ring‎, ‎and let i subset j be two ideals of a‎. ‎let m be a finitely generated a-module‎. ‎brodmann proved that the function n mapsto mathrm{depth}_{j}(frac{m}{i^{n}m}) is constant for large n‎.‎in this paper‎, ‎we consider a filtration phi = (m_n)_{n in mathbb{n}} of m and a filtration f = (i_n)_{n in mathbb{n}} of a‎. ‎generalizing brodmann's result‎, ‎we first show that the function n mapsto mathrm{depth}_{j}(frac{m}{m_{n}}) is constant for large n of value mathrm{depth}_{j}(f‎, ‎m)‎, ‎provided that phi is f-good and f is strongly noetherian‎. ‎secondly‎, ‎we establish the inequality‎ ‎gamma_{j}(f‎, ‎m) leq dim_a(m)‎ - ‎mathrm{depth}_{j}(f‎, ‎m)‎, ‎where gamma_{j}(f‎, ‎m) denotes the analytic spread of f at j with respect to m‎.