On upper bounds for asymptotic ideal-grade

Let $I$ and $J$ be ideals in a Noetherian ring $R$ and let $x_1,\dots ,x_n$ be nonunits in $R$. Then $x_1,\dots ,x_n$ is said to be an asymptotic sequence over $I$ if $(I,(x_1,\dots ,x_n))R\ne R$ and if for all $1\le i\le n$, $x_i$ is not in any associated prime of the integral closure $\overline{{(I_{i-1})}^m}$ of ${(I_{i-1})}^m={(I,(x_1,\dots ,x_{i-1}))}^{m}R$, where $m\in \mathbb{N}$ is very large. Let ${agd}_I(J)$ be the maximum number of elements in $J$ which form an asymptotic sequence over $I$. It is characterized when ${agd}_I(J)$ is equal to: (i) $\ell (J)$, the analytic spread of $J$, when $R$ is local; (ii) $ht(I+J)-agd(I)$, where $agd(I)$ is the maximum number of elements in $I$ which form an asymptotic sequence over $(0)R$, and several consequences of these characterizations are given. Finally, if $R$ is local with maximal ideal $𝔪$ then we reprove a known upper bound for ${agd}_I(𝔪)$.