  Recently, Okounkov, Lazarsfeld and Mustata, and Kaveh and Khovanskii have
shown that the growth of a graded linear series on a projective variety over an
algebraically closed field is asymptotic to a polynomial. We give a complete
description of the possible asymptotic growth of graded linear series on
projective schemes over a perfect field. If the scheme is reduced, then the
growth is polynomial like, but the growth can be very complex on nonreduced
schemes.
  We also give an example of a graded family of m-primary ideals {I_n} in a
nonreduced d-dimensional local ring R, such that the length of R/I_n divided by
n^d does not have a limit, even when restricted to any arithmetic sequence.
