On asymptotic depth of integral closure filtration and an application

Let $(A,\mathfrak{m})$ be an analytically unramified formally equidimensional Noetherian local ring with $depthA\ge 2$. Let $I$ be an $\mathfrak{m}$-primary ideal and set $I^{\ast }$ to be the integral closure of $I$. Set $G^{\ast }(I)={⨁}_{n\ge 0}(I^n{)}^{\ast }\mathrm{/}(I^{n+1}{)}^{\ast }$ be the associated graded ring of the integral closure filtration of $I$. We prove that $\ depth
G^(I^n) \geq 2$forall$n \gg 0$\mathrm{.}Asanapplicationweprovethatif$A$ is also an excellent normal domain containing an algebraically closed field isomorphic to $A\mathrm{/}\text{m}$ then there exists $s_0$ such that for all $s\ge s_0$ and $J$ is an integrally closed ideal \emph{strictly} containing $({\mathfrak{m}}^s{)}^{\ast }$ then we have a strict inequality $\mu(J) < \mu((\mathfrak{m}^s)^)$(here$\mu(J)$isthenumberofminimalgeneratorsof$J$).