Hilbert-Kunz density function for graded domains

We prove the existence of HK density function for a pair $(R,I)$, where $R$ is a $\mathbb{N}$-graded domain of finite type over a perfect field and $I\subset R$ is a graded ideal of finite colength. This generalizes our earlier result where one proves the existence of such a function for a pair $(R,I)$, where, in addition $R$ is standard graded. As one of the consequences we show that if $G$ is a finite group scheme acting linearly on a polynomial ring $R$ of dimension $d$ then the HK density function $f_{R^G,{\mathbf{m}}_G}$, of the pair $(R^G,{\mathbf{m}}_G)$, is a piecewise polynomial function of degree $d-1$. We also compute the HK density functions for $(R^G,{\mathbf{m}}_G)$, where $G\subset S{L}_2(k)$ is a finite group acting linearly on the ring $k[X,Y]$.