Asymptotic $\mathrm{v}$-number of graded families of ideals and the Newton-Okounkov region

In this paper, we prove that for Noetherian graded families $\mathcal{I} = \{I_k\}_{k \ge 0}$ of homogeneous ideals in a Noetherian $\mathbb{N}$-graded Noetherian domain, $\lim\limits_{k \to \infty} \frac{\mathrm{v}(I_k)}{k}$ exists, and is given by $\frac{α(I_r)}{r}$ for some $r \ge 1$, where $α(I)