Growth of multiplicities of graded families of ideals

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d>0$. Let $I_{\bullet }=\{I_n{\}}_{n\in \mathbb{N}}$ be a graded family of $\mathfrak{m}$-primary ideals in $R$. We examine how far off from a polynomial can the length function ${\mathrm{\ell }}_R(R\mathrm{/}{I}_n)$ be asymptotically. More specifically, we show that there exists a constant $\gamma >0$ such that for all $n\ge 0$, $${\mathrm{\ell }}_R(R\mathrm{/}{I}_{n+1})-{\mathrm{\ell }}_R(R\mathrm{/}{I}_n)<\gamma n^{d-1}\mathrm{.}$$