Multigraded Fujita Approximation

The original Fujita approximation theorem states that the volume of a big divisor $D$ on a projective variety $X$ can always be approximated arbitrarily closely by the self-intersection number of an ample divisor on a birational modification of $X$. One can also formulate it in terms of graded linear series as follows: let $W_{\bullet }=\{W_k\}$ be the complete graded linear series associated to a big divisor $D$: $$W_k=H^0(X,{\mathcal{O}}_X(kD))\mathrm{.}$$ For each fixed positive integer $p$, define $W_{\bullet }^{(p)}$ to be the graded linear subseries of $W_{\bullet }$ generated by $W_p$: W^{(p)}_{m}={cases} 0, &\text{if $p\nmid m$;} \mathrm{Image} \big(S^k W_p \rightarrow W_{kp} \big), &\text{if $m=kp$.} {cases} Then the volume of $W_{\bullet }^{(p)}$ approaches the volume of $W_{\bullet }$ as $p\to \mathrm{\infty }$. We will show that, under this formulation, the Fujita approximation theorem can be generalized to the case of multigraded linear series.