Asymptotic periodicity of grade associated to multigraded modules

Let $R$ be a Noetherian $\mathbb{N}^r$-graded ring generated in degrees
${\bf d}_1, \dots , {\bf d}_r$ which are linearly independent vectors over $\mathbb{R}$, and let $\mathfrak{a}$ be an ideal in $R_{\bf0}$. In this paper, we investigate the asymptotic behavior of the grade of the ideal $\mathfrak{a}$ on the homogeneous components $M_{\bf n}$ of a finitely generated $\mathbb{Z}^r$-graded $R$-module $M$ and show that the periodicity occurs in a cone.