  We give simple necessary and sufficient conditions on projective schemes over
a field k for asymptotic limits of the growth of all graded linear series of a
fixed Kodaira-Iitaka dimension to exist. We also give necessary and sufficient
conditions on a local ring R of dimension d for the limit
\lim_{i\rightarrow\infty}\frac{\ell_R(R/I_n)}{n^d} to exist whenever \{I_n\} is
a graded family of m_R-primary ideals. We give a number of applications.
